Cyprus used to be generally regarded as a Greek island, as the quotations below indicate. However, from the 1950s, when the Greek Cypriots intensified their campaign for liberation from British rule and union with Greece, Britain and Turkey began a partially successful campaign to change this perception.
In order to justify keeping possession of the island, Britain, labouring under the erroneous ideas that enosis was being promoted by a handful of fanatical clergymen and a few self-seeking politicians, and that there was no widespread public support for union with Greece, embarked on a campaign to de-Hellenise perceptions of Cyprus both within and without the island. This involved using Turkey and the Turkish Cypriots to counter Greek Cypriot claims, and disseminating the ideas that Cyprus has never been Greek; and the Cypriots are not really Greek, they just think they are.
In the mid-1950s Turkey began promoting its own claim to all of the island (or at least half if we can’t have it all). This was based on geographical proximity, strategic and military “imperatives”, and the fact that the island had been under Ottoman rule before the British took it over. The presence of a Turkish Cypriot minority was an added bonus. The Turks too made much of the idea that “Cyprus has never been Greek”, which for them meant that it had never been ruled by Greece, whereas it had been ruled by Turkey and should be given back. This was nonsense. The Ottoman Empire was not Turkey, but an entirely distinct political entity which was extinguished at the end of the First World War.
None of these British and Turkish arguments should be taken seriously.
The claims made by Turkey were encouraged by the British, who, for example, took no action against the establishment of a Turkish Cypriot political party in July 1955 with the provocative name “Cyprus Is Turkish”.
I have organised the quotations below chronologically. It should be noted that from the 1950s many of the quotations are responses to the absurd ideas being put about by representatives of the British Government.
Particularly interesting is the quotation from Bitter Lemons by Lawrence Durrell, a self-confessed Tory
and imperialist, who took charge of the British Colonial Government’s Press and Information Office in Cyprus in 1955. The quotation
represents a moment of clarity in a book bedevilled by the erroneous ideas outlined in paragraph 2 above.
“Cyprus surpasses every other Greek island in the number of natives illustrious for their birth, dignity, learning and saintliness.” 
“Speaking at Penicuik, Scotland, on 25 March 1880, while still in opposition, he emphasized his desire that, sooner or later, Cyprus should be united with Greece because, apart from other considerations, ‘the bulk of the people are Greek.’ On 17 December of the same year he suggested to Glanville (a leading Liberal politician) that Cyprus should be handed to Greece by Britain and Turkey ‘in sovereignty not in mere occupation.’ He was of the same opinion about Crete which he wished to be ceded to Greece. He had repeatedly stated that Crete’s union with Greece was not only desirable but essential.” 
“Writing to the Duke of Westminster a long exposition of his half century’s experience of the Eastern Question, he expressed his hope of having the satisfaction ‘to see the population of that Hellenic island placed by a friendly arrangement in organic union with their brethren of the Kingdom [of Greece] and of Crete.’” 
“Now that Serbia has been attacked by Bulgaria, if Greece is willing to come to her aid, His Majesty’s Government is ready to cede to Greece the island of Cyprus. If Greece joins the Allies for all purposes, she will naturally participate in the advantages secured at the end of the war, but the offer of Cyprus is made by His Majesty’s Government independently of this consideration, and on the sole condition that Greece gives Serbia her immediate and complete support with her army.” 
“Several months after the 1931 uprising, Professor Arnold Toynbee, writing in the New Statesman and Nation on 23 April 1932, as in the Economist earlier, urged the cession of Cyprus to Greece. Toynbee’s published opinion caused the Colonial Office’s displeasure to such an extent that it called it ‘highly disparaging to British rule in Cyprus; tendentious, misleading, one-sided and insidiously pro-Hellene.’ The Colonial Office’s reaction was proportionate to the weight it attributed to Toynbee’s opinion. It was widely acknowledged that Professor Toynbee was totally aware of what was understood as the ‘Eastern Question’ and that he knew quite well all its parameters relating to Greece and Turkey, since it was he who, as a staffer of the Political Intelligence Department (P.I.D.), shouldered, along with Harold Nicolson and Allen Leeper, the greater bulk of the work on preparing Britain’s policy towards those two countries in the face of the Paris Peace Conference (1918-1919).” 
“... the Greek colonists Hellenized Cyprus as early as the fourteenth century B.C.: the Phoenicians did not arrive until the eleventh century, and then only to occupy two coast stations, Kition and Lapithos, for trading with the indigenous population. Thirty-three centuries of occupation constitute a claim much longer and far more continuous than the claim of Israel to Palestine. But even if Greek origin could not be proved for three thousand years, and had to be post-dated to the comparative modernity of Alexander the Great or to the Byzantine Empire; even if the Athenian of to-day is apt to distinguish Cypriots from other Greeks by the opprobrious epithet of βους Κύπριος—the equivalent of the German Ochsenkopf the Greekness of Cypriots is in my opinion indisputable. Nationalism is more, is other, is greater than pigmentations or cephalic indices. A man is of the race of which he passionately feels himself to be. No sensible person will deny that the Cypriot is Greek-speaking, Greek-thinking, Greek-feeling, Greek, just as much as the French Canadian is French-speaking, French-thinking, French-feeling, French. Both are equally sensitive upon this matter of race...” 
“The noble Earl, Lord Swinton, said that Cyprus was never a Greek possession. Cyprus has a history of some 3,000 years. It is a Greek island; it was a free Greek island. It possessed itself for many years as a free Greek island with the free City States. It was a portion of Magna Graecia, the great classical Greece. It was part of the Byzantine Empire. Then there were certain vagaries of control after the collapse of the Byzantine Empire. For 300 years it was under Turkish rule, and for 78 years it has been under British rule. But all the time the people of Cyprus have been a Greek people. They have retained the Greek character and used the Greek language. And all the time there has been, sometimes strong, sometimes weak, but in these days stronger because of the pace of the times, the historic urge to fulfil their destiny in association with Greece.” 
“For more than thirty years I have believed that Cyprus, like other parts of the British Commonwealth, was destined, with British help, British goodwill and British guidance to achieve self-government, and thereafter in exercise of self-determination, by the will of its people and by the will of ours, to become a happy and prosperous part of Greece.” [....]
“The Tory Party still use, and I am afraid they still believe, those empty legends that Cyprus has never been Greek...”
“‘Cyprus has never been Greek?’ I had that put through my letter box in a Tory leaflet the other day. ‘Cyprus has never been Greek?’ It has never been anything else since recorded history began. It has had conquerors, as Britain has had conquerors. The French were in Cyprus before the Turks, and longer; but its Greek traditions and culture have never been subdued. It has never been part of the kingdom of modern Greece, no; nor were the Ionian Islands; nor Crete, nor Mytilene; nor the Dodecanese, until they were given to Greece by Britain or the Powers.” [....]
“I know that hon. Members opposite find it hard to think that the Cypriots are not happier under us. They do not understand the passionate pride of the Cypriots in being Greek; in sharing the ancient glory and what they think the modern glory of the Greeks. In listening to the Athens Radio—which I abhor like other hon. Members—we have half forgotten what kind of qualities the modern Greeks still have.”
“We have forgotten that in 1941 Greeks were killed in the streets of Athens by Germans because they refused to allow British prisoners of war to have the humiliation of being made to sweep the streets. Greek lives were given to save our prisoners from shame. For twenty centuries men have talked about three hundred Spartans who combed their hair before they fought the barbarians at Thermopylae, knowing that they were all to die. In Macedonia in 1941 there were three Thermopylaes. With the road to Salonica cut behind them, outnumbered ten to one, with no modern arms, 500 Greeks held a fort at Perichori until not one man remained alive. And two other garrisons did the same...” 
“I could not help reflecting wryly that had we been honest enough to admit the Greek nature of Cyprus at the beginning, it might never have been necessary to abandon the island or to fight for it. Now it was too late.” 
Sir, —The leader of the “Cyprus is Turkish” association of Cyprus is reported on July 19 to have emphasized that “Turkish Cypriots, having the full support of the Turkish Government, were more than ever insistent on partition of the island.” But surely if they believe that “Cyprus is Turkish,” partition should be an utterly intolerable prospect to them, as it is to the Greeks? It might be worth reminding them, in their own interests, that the last time a comparable situation arose, it was settled by the Judgment of Solomon. (“Cyprus”, The Times, 25 July 1957, p. 11.)
“The casual visitor can travel from end to end of the island completely unaware that Turks existed there. Very occasionally a mosque can be seen or a broken-down advertisement in Turkish. But the overwhelming impression is Greek in towns and villages—churches, roads, advertisements, place-names.” 
“Cyprus is and always has been overwhelmingly Greek...” 
“Nothing could be more absurd than the fashionable slogan of the 1950s, that ‘Cyprus never has been Greek’. What else has it ever been?”