The Portugal international, aged 28, spent last season on loan at his former club Sporting Lisbon.
Towards the end of his loan spell, Nani said he would be reporting for pre-season duty with United – but it now appears he might not get to that stage.
He is said to be on the verge of signing a four-year contract worth £3m-per-season with the Turkish Super Lig runners-up.
Nani was touted as the new Cristiano Ronaldo when he followed his compatriot from Sporting to United in a £17m in 2007.
He has played 230 games for the club in all competitions, scoring 40 goals, but has never hit the heights of his national team colleague.
United boss Louis van Gaal shipped him out on loan soon after taking charge at Old Trafford last summer.
Although he impressed at Sporting last season, it never seemed likely that Nani would force his way into Van Gaal's plans.
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