The newspaper report claims that Fenerbahçe have found sponsors that will finance the deal if Van Persie were to accept terms with the club.
United boss Louis van Gaal has been disappointed with his strikers this season and could make a number of changes in the summer.
“We have to improve. Improve in creating chances? No. Improve in finishing,” Van Gaal was quoted as saying by the Express after the 1-0 loss against West Brom on the weekend.
Van Persie missed a penalty in the loss against the Baggies and has scored his lowest goal tally since the 2009-10 league campaign.
The Netherlands international does however, want to stay at United next season according to the Daily Mail.
The 31-year-old is reported to have accepted he may be demoted to a more peripheral role.
Van Persie will enter the final year of his United contract next season but the club have a policy of offering one-year deals to players over the age of 30.