Turkish outlet Fanatik reports that Rashford has become the club's "latest target" after Fenerbahce completed a deal to sign fellow former United forward Mason Greenwood from Marseille.
According to the report, Fenerbahce's football director Cihan Kamer and sporting director Oguz Cetin travelled from Marseille to Manchester after finalising the Greenwood deal, with official talks over Rashford, 28, now set to begin.
Rashford is no longer part of United's first-team plans after returning from a season-long loan at Barcelona, where he helped the Catalans win LaLiga. Manchester United had been hoping Barcelona would make the move permanent for a reported £26m, but the Spanish club instead signed Anthony Gordon from Newcastle United and are closing in on a deal for Borussia Dortmund's Karim Adeyemi.
That has left both United and Rashford in an awkward position, with the England international currently away at the World Cup and due back at Carrington for pre-season. Fenerbahce could offer a route out, with Fanatik reporting the Turkish club are "keen to reach an agreement" as soon as possible.
However, the move is not as advanced as some reports suggest. Transfer journalist Fabrizio Romano said this week that there is "nothing" between Rashford and Fenerbahce at this stage, with no negotiations or conversations having taken place. United boss Michael Carrick is instead said to be planning to assess Rashford in pre-season training before deciding on his future at the club.
Reports in Turkey have also floated a fee in the region of £40m, a figure that could tempt United's hierarchy given Rashford's contract situation and his uncertain role within Carrick's squad. Alejandro Garnacho and Jadon Sancho have both already left Old Trafford, leaving Rashford and Amad Diallo as United's only recognised senior wingers heading into the new season.
Should he depart, United are understood to be exploring alternatives on the flank, including West Ham's Crysencio Summerville, who impressed for the Netherlands at this summer's World Cup. For now, though, Rashford's immediate focus remains on England's World Cup campaign, with his club future set to be resolved only once the tournament finishes and pre-season begins in earnest.