The report claims that the Black Eagles are plotting a move for the Germany international midfielder following reports that United have put the him up for sale.
Besiktas are keen on strengthening their midfield with players that have Champions League experience.
The Black Eagles currently lack depth in midfield and with Champions League football looming the club have made it a priority to bring in reinforcements.
The Daily Record, reports that United boss Jose Mourinho has deemed Schweinsteiger, Marcos Rojo, Daley Blind and Juan Mata as he prepares to rearrange his squad for next season.
Besiktas could however, face competition from Fenerbahce who are also reported to be in the race for the experienced midfielder.
French giants PSG are also reported to be interested in the midfielder according to the Sun.
Schweinsteiger has won eight Bundesliga titles and the Champions League over an illustrious career.