The German international only moved to Old Trafford last summer in a £6.5million deal from Bayern Munich, where he lifted eight Bundesliga titles and the 2013 Champions League title.
But injuries limited the 31-year-old to just 13 starts in the Premier League last season and only 31 outings across all competitions.
The Daily Record claim Jose Mourinho is looking to trim the Red Devils’ squad after making three summer signings and Schweinsteiger is one of the players set for the chop.
And according to LigTV, Besiktas have already held talks with United regarding a potential season-long loan move for the veteran midfielder.
The Super Lig champions, who won last season’s title with five points to spare, want a high-profile signing of Schweinsteiger’s calibre to mark their move to a new stadium.