The report claims that the Yellow Canaries have had the Germany international on their shortlist since last season but recent reports suggesting United have put the central midfielder up for sale has raised the possibility of a summer offer.
The Istanbul giants are reported to have made an enquiry to the Red Devils for the 31-year-old midfielder.
The Daily Record, reports that United’s boss Jose Mourinho is keen on trimming his squad and that Schweinsteiger, Marcos Rojo, Daley Blind and Juan Mata are currently deemed surplus to requirements.
Fenerbahce have already completed three defensive transfers this summer and will focus on bolstering the midfield next.
Schweinsteiger would become the third United player to join the Istanbul giants within the span on a season if he were to join.
The Yellow Canaries finished last season as runner-up and will play Champions League football next season.