The 31-year-old scored 30 goals for United as they won the league in 2012-13 following his Â£24million move from Arsenal.
His impact since then, however, has been muted due to a series of niggling injuries, and he was limited to 10 goals in 29 appearances under compatriot Louis van Gaal last season.
On Thursday, Fenerbahce announced via Twitter: "Fenerbahce has begun to transfer talks with Robin Van Persie and his club Manchester United."
Fenerbahce signed Van Persie's former United team-mate Nani on Monday for a reported Â£4.25million.
If Van Persie is sold, United would have only Wayne Rooney, James Wilson and the injured Javier Hernandez - out with a broken collarbone - in their squad as recognised strikers following the exit of on-loan Falcao at the end of the season.
Source : PA