"Bastian wants to do something new at the end of his career," Rummenigge told a news conference.
"The colleague from Manchester United got in touch with me. We have agreed on a transfer.
"I regret that because Bastian was an extremely deserving player for us. He has done great things.
"I can only wish all the best to Bastian. It will be an interesting task for him."
United confirmed they had reached an agreement to sign the 30-year-old subject to a medical and a deal on personal terms with the 13-times Premier League champions.
A World Cup-winner with Germany in Brazil last year, Schweinsteiger came through the youth ranks at Bayern and has spent 13 seasons with the Bavarians, winning the Bundesliga title eight times.
During a glittering career, Schweinsteiger, who made 536 appearances for the club since his debut in 2002, also won seven German Cups and the Champions League in 2013.
Schweinsteiger will be the first German to play for United and their second signing of the close-season following the arrival of Netherlands winger Memphis Depay from PSV Eindhoven for a fee of around 30 million euros ($33.5 million).
($1 = 0.8960 euros).
Image: Bastian Schweinsteiger in action for Germany.
Photograph: Matthias Hangst/Getty Images
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