Rummenigge said the German champions tried to convince the Germany captain to stay at the Allianz Arena but the 30-year-old felt the need for a new challenge.
He would not confirm the transfer fee though it is believed Bayern want at least £15m for a midfielder who has given them stellar service in the first team over the past 13 years.
United’s medical staff are awaiting Schweinsteiger’s arrival for a medical at Carrington on Saturday afternoon. He will become the first ever German to play for the Old Trafford club.
Speaking at a press conference to unveil Bayern’s summer signings, Rummenigge also found himself announcing a high profile departure.
He said: ‘Bastian wants to do something new at the end of his career. He asked that we meet his wishes.
‘My colleagues at Manchester United have been in touch. We have reached an agreement over a transfer.
”It’s regrettable because Bastian meant a great deal to the club. He achieved great things.
‘We say goodbye on great terms. ‘I can only wish Bastian all the best. It will be an interesting new challenge for him. I am convinced that he will play well there.’
Reflecting on the player’s legacy at Bayern, Rummenigge added:
‘He has done great things here, he has won everything.
‘I suggest to him that we will give him a great farewell match in Munich when he finishes his career, whenever that will be. And when he stops, we will talk about whether there is a second career for him here at Bayern.’
Manchester United confirmed the signing via their official Twitter feed shortly after Rummenigge had finished speaking.
A statement on United’s official website said:
‘Manchester United is delighted to announce it has reached agreement with Bayern Munich to sign midfielder Bastian Schweinsteiger, subject to a medical and personal terms.’
Germany coach Joachim Low said the midfielder would thrive at United.
‘Basti is an absolute leader and world-class player who can put his stamp on any team, including of course Manchester United,’ he said on www.dfb.de.
‘He knows Louis van Gaal, who wanted him unconditionally. I know from him that he still has big ambitions and goals.
‘He will approach his new challenge in England like we all know him – with dedication and highly-motivated.
‘For Bayern and the Bundesliga his departure is without question a loss.’
The German newspaper Bild reported on Friday night that Schweinsteiger had agreed a three-year contract worth £7.2m-a-year with United and was on standby to fly to England to complete the formalities of a medical and to sign the contract.
The move to Manchester ends a trophy-laden 17-year association with the Bavarian club, for whom Schweinsteiger made a total of 500 first-team appearances and scored 67 goals.
A German international of 111 caps, he was part of the team that lifted the World Cup in Brazil last year.
This is a huge signing for van Gaal and Manchester United. With the retirement of Sir Alex Ferguson and the departure of veterans like Vidic, Evra, Ferdinand, Scholes, Fletcher and Giggs, the United dressing room lost that veteran leadership. Players who knew what it takes to win trophies. Schweinsteiger will bring that experience and leadership back to United.