Sources in Germany confirmed reports in a German magazine that claim a deal is "close to being done".
United will pay £18m for Hargreaves, 26, and set a record fee for a player from the Bundesliga.
Hargreaves, a long-standing target of United boss Sir Alex Ferguson, will sign a four-year deal with the newly-crowned Premiership champions.
Bayern have promised to spend big following their failure to qualify for next season's Champions League, with chairman Karl-Heinz Rummenigge ready to carry out "brutal and ruthless" changes to the team.
The German giants have struggled to keep pace with the spending power of Europe's elite, and Rummenigge and advisory board chairman Franz Beckenbauer hinted the sale of players like Hargreaves would fund their rebuilding process.
Beckenbauer said: "We'll have to invest a lot of money if we want to be at the top next year and succeed at home and abroad.
Rummenigge added: "All I can say is that there will be changes in every area of the team, some players will come and some players will go."
According to reports, Bayern will use the proceeds of Hargreaves' sale to fund a bid for Chelsea winger Arjen Robben.
Hargreaves missed Bayern's 5-2 win over Mainz because of a knee injury which means he could have already played his last game for the club after 10 years.
He has played a total of 145 league games and won the Champions League with the Bavarian side in 2001.
Beckenbauer, speaking recently on German television, said: "If it is right there is an offer of 25-30m euros (£17-20m), then I would seriously consider it.
"I think the board also think this way to satisfy the needs of the player."
Hargreaves has been a long-standing United target and the player has expressed a desire to play in England.
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Question:
What difference do you think Hargreaves would have made to the Man Utd team yesterday, if he had played instead of Fletcher?