The 30-year-old is believed to have told Bayern he does not want to extend his contract, which runs to 2016.
The clubs are still to agree a fee, but United are prepared to give the World Cup winner a massive increase on his salary.
Schweinsteiger has made 536 appearances for Bayern since his debut in 2002.
Should the German arrive, it would be a significant statement of intent from the Old Trafford club, similar to the one they made last summer with the arrivals of Angel Di Maria from Real Madrid for £59.7m and the loan signing of Radamel Falcao from Monaco.
While Falcao has since moved to Chelsea after scoring only four goals for United and Di Maria was one of the players manager Louis van Gaal said on Friday “had problems” during his first season, their presence did underline a desire to reclaim the club’s previous status in the Premier League.
Along with Real Madrid defender Sergio Ramos, Schweinsteiger was identified earlier this summer as a high-profile player United were interested in.
While the chances of Ramos ending up at Old Trafford are tied up in the future of goalkeeper David De Gea – a target for Real – Schweinsteiger is expected to end up as a United player.
The club have made only one signing so far this summer, with £31m winger Memphis Depay arriving from Dutch club PSV Eindhoven.
However, Italian defender Matteo Darmian is flying to England for a medical before completing a £13m move from Torino.
It is anticipated he will travel with United for their pre-season tour of the United States, which begins on Monday.
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