Red Devils boss Louis van Gaal believes he can convince his former player to quit the Allianz Arena in favour of Old Trafford, with United executive vice-chairman and chief transfer negotiator Ed Woodward being handed the responsibility to complete the deal.
German international Schweinsteiger turns 31 in August and has only a year left on his current contract with German champions Bayern.
He is believed to be interested in linking up with Van Gaal again, with the Dutchman having managed him when in charge at the Bavarian club between 2009 and 2011.
Under Van Gaal, Schweinsteiger won the Bundesliga title and the DFB-Pokal, both during the 2009/10 season. The German has since gone on to win the Champions League and the FIFA World Club Cup, and has amassed a total of 500 appearances in all competitions for Bayern.