The Turkish newspaper AMK has reported that the Istanbul Giants are looking to strengthen in midfield ahead of next season and are preparing a move for the Germany international.
Schweinsteiger has been plagued with injury troubles since joining United from Bayern.
Van Persie and Nani are both in great form for second placed Fenerbahce this season having scored, 15 and 10 goals, respectively.
Schweinsteiger has two assists and one goal for United in 31 games since joining in the summer. The 31-year-old is currently undergoing treatment for a ligament injury.
United take on Everton in the Premier League this weekend as they look to secure a forth place finish.