The Red Devils have already had two bids for the Barcelona midfielder rejected and are ready to make a third offer that will reach £40m after add-ons.
According to The Mirror, they will move for Bayern Munich's Schweinsteiger if that bid fails.
The German international midfielder, who has been linked with United in the past, may struggle to hold down a regular place at his club this season under Pep Guardiola's new system.
It is believed that Schweinsteiger will command a fee of around £25m despite the fact that he is 29. If he is available, then the likes of Arsenal and Chelsea could also make a move.
Other options for United are Everton's Marouane Fellaini and Luka Modric of Real Madrid, although both clubs are very reluctant to sell.