I doubt they or the rest of the NATO alliance would countenance a Russian base on a NATO member's soil
There is no strategic sense at all to that for anyone
NATO membership is looking pretty useful to Turkey about now, given the deterioration of Syria and Iran's future ICBMs
Now, a deal between Cyprus and Russia for the sale ...... err, “development” .... of Cyprus’ supposed offshore natural gas fields - is potential Russian domination of the Central Med by another means
But very few countries actually recognize Turkish claims on the island. I would be surprised if NATO would, seeing as how Greece, in particular wouldn’t care for it.