Louis van Gaal, who will take charge of United following his World Cup commitments with Holland, is expected to oversee a summer overhaul at the club and has identified the Germany international as a top target.
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United is set to make a formal offer for the 29-year-old in the coming days as executive vice-chairman Ed Woodward, who was heavily criticized for the club's perceived failure in the transfer window last season, gets to work on Van Gaal's list of potential signings.
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Schweinsteiger is seen as a Bayern Munich icon but Goal understands that United is increasingly confident that the midfielder could be persuaded to make the switch to Old Trafford and be reunited with former boss Van Gaal, following third-party conversations.
Although Pep Guardiola's side would be reluctant to part with one of its star players, at 29 it could be a case of now or never if the Bavarians are to cash in for a significant fee.
Van Gaal had also identified Thomas Muller, a player he brought through the ranks at Bayern while in charge there, as a transfer target, but the playmaker ended speculation about his future by signing a new contract at the Allianz Arena that ends any possibility of a summer departure.
Forward Muller, 24, had recently cast doubt on his future, saying: "I have now come to the point where I insist to know that I am part of the starting 11 at Bayern Munich. I have also demanded to know if the club really want to go forward with me in the next few seasons.
“I am no longer the type who says, ‘OK then, today I will play in this role, next match in that role. Just move me about and I will keep my mouth shut’. No, those days are gone.’’
But United officials have long suspected that Muller's preference was to stay at Bayern, and confirmation that he has agreed a new deal will come as little surprise at Old Trafford.