The Black Eagles are targeting a central midfielder and have been keeping tabs on the Germany international.
Per the source, Besiktas want the 31-year-old on loan but would be willing to enter talks over a permanent transfer for the right price.
PSG are also in the race for the experienced Germany international according to the Sun.
The report claims thatUnited are willing to let Schweinsteiger leave but he wants all of the £14 million he is set to receive over the next two-seasons if he does part ways.
Beiktas are currently monitored by FFP and must show higher revenues than spending as part of the terms of their current agreement.
Besiktas will compete in the Champions League next season are are keen on adding an experienced midfielder who has previously played in the competition.
Schweinsteiger has not only played Champions League football he lifted the trophy at Bayern Munich. The central midfielder has won just about everything there is to win at club and international level including the World Cup.