British agents Mustafa Kemal and T.E. Lawrence together - They did know each other

Source: Lawrence of Arabia, by J Wilson
page 558:

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Kemal was held briefly by the Arabs and interrogated by Lawrence before being released. Kemal had been corresponding with Faisal for several months, and the Arab Nationalists saw his Pan-Turk party as a potential ally. As a prisoner, he would have been in no position to further their cause. Some years after the war, Lawrence told a Foreign Office official that "by a curious accident he was able, in September 1918, to have several conversations with Mustafa Kemal Pasa."

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Lawrence recalled that they had talked, among other things, about Turkish war aims and the aspirations of the Pan-Turk party. His statement gave the gist of these conversations in considerable detail. Kemal had told him that Turkey's real interests lay to the east. They had entered the war primarily to gain territory in Persia, Muslim Trans-Caucasia, and so on. He had confirmed that the Pan-Turks were not interested in the Arab Provinces: Palestine, Syria, Mesopotamia were not only valueless in the Pan-Turkish scheme of things(except in certain strategical aspects relating to the war) but would be positive dangers and encumbrances if they remained in Turkish possession. The Pan-Turks, he declared, would lose them without a regret; they would even be glad to be rid of them.

Although Kemal was a high-ranking enemy officer, the Arab leaders saw him as their best hope for future relations with Turkey, and with Lawrence's agreement he was released.

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Both Faisal and Lawrence were later to be staunch supporters of Kemal during his struggle for control in Turkey and for international recognition.