The Turkish club’s director Guiliano Terraneo travelled to London to discuss a potential deal with United chiefs for the 31-year-old Netherlands striker, who has one year left on his contract at Old Trafford.
Former United winger Nani completed a move to Fenerbahce earlier this week worth a reported £4.5m and the Portugal international is keen for Van Persie to join him in Turkey.
“I haven’t had the chance to talk with Robin van Persie yet,” Nani said. “But if he comes to Fenerbahce, I am sure that he will be very successful.
“I know Robin very well and it would be the right decision if Fenerbahce sign him.”
Van Persie’s potential exit could leave United short of recognised strikers heading into the new season.
The club elected not to pick up the option to buy Radamel Falcao – who subsequently joined Chelsea – from Monaco after the Colombian’s disappointing form last season.
Javier Hernandez is set to return to the club following a season on loan at Real Madrid but faces a spell on the sidelines after suffering a suspected broken collarbone while playing for Mexico against Honduras.
Captain Wayne Rooney and teenager James Wilson are Louis van Gaal’s only two other striking options.