Rashford is back at Manchester United after a successful loan spell at Barcelona, with manager Hansi Flick admitting he will miss the England international after it was announced the Catalan club would not be signing him permanently.
Rashford scored 14 goals and provided 11 assists in 49 appearances for Barcelona last season, but the Spanish champions chose to spend their attacking budget on Anthony Gordon instead, signing the winger from Newcastle for around £70m rather than triggering their option to make Rashford's move permanent.
Rashford is on a three-man shortlist for the Turkish giants that also includes AC Milan winger Rafael Leão and Crystal Palace forward Ismaila Sarr. Fenerbahce have already held talks with Leão but have so far failed to agree a deal. They have also contacted the representatives of Sarr, who is represented by the same agent as Chelsea's Nicolas Jackson.
Rashford's representatives have also been approached over a prospective move, as Fenerbahce look to back new manager Ismail Kartal with a squad capable of challenging rivals Galatasaray for the Süper Lig title.
Fresh from England's third-place finish at the World Cup, Rashford will take his time before deciding on his next move, though sources say Fenerbahce are prepared to break the bank for him if he wants to experience the passion of Turkish football.
Sarr is another player Fenerbahce admire following a terrific campaign in which he scored 21 goals and helped Crystal Palace win the Europa Conference League, before proving himself on the biggest stage by scoring four goals at the World Cup, making him Senegal's all-time leading scorer in the tournament's history.
Leão, meanwhile, has been the subject of persistent Milan exit talk for over a year without a move ever materialising.