Bayern ceo Karl-Heinz Rummenigge said he met Schweinsteiger on Friday and the midfielder said he had decided to go.
“We have responded to this request,” Rummenigge said. “My colleague from Manchester United has contacted me. We have agreed on a transfer.”
Rummenigge added: “We tried to convince him to stay with Bayern Munich. But I also understand that a player who has spent 17 years at Bayern Munich, at the end of his career still wants to gain a new experience.”
Describing him as “an icon, a legend”, the chairman also said he hoped to see Schweinsteiger back at the club in the future.
“He has done great things here, he has won everything,” he said. “I suggest to him that we will give him a great farewell match in Munich when he finishes his career, whenever that will be. When he stops, we will talk about whether there is a second career for him here at Bayern.”
Schweinsteiger has won the Bundesliga eight times, lifted the Champions League with Bayern in 2013 and was part of Germany’s World Cup winning squad last summer.
After winning 111 caps for Germany, he is now poised to become United’s third signing of the summer following the £25million arrival of Memphis Depay from PSV Eindhoven and Torino right-back Matteo Darmian.
The defender also has 13 international caps for Italy and was part of their World Cup squad last summer in Brazil. He has represented his country at every youth level from Under-17 to Under-21.
Darmian said: “It is a dream come true to be joining Manchester United. I have loved the time that I have spent at Torino and I would like to thank the club and the fans for everything they have done for me.
“I am sure they can understand when the opportunity to join the biggest club in world came along it was something I had to do. To work with Louis van Gaal, a manager that has achieved so much in football is such an exciting prospect.”
Darmian, born in Legnano, Milan, has agreed a contract that will keep him at Old Trafford until 2019 with the option to extend for a further year. The fee is believed to be around £13million.
Van Gaal added: “Matteo is a right-sided full back with the versatility of also being able to play on the left. He is a strong defender and has the ability to go forward in the attacking positions which is a fantastic attribute to have and much needed in the fast rhythm of the Premier League.
“Matteo will be a great addition to the team. I am delighted to welcome Matteo to the club and look forward to working with him in the coming days.”
Darmian was at Milan from 2001 to 2010 before being sold to Palermo, who are due roughly five per cent of the player’s transfer fee, which amounts to around £287,000.
United’s business is far from complete. They are still considering pressing for Southampton’s France midfielder Morgan Schneiderlin and continue to to Real Madrid that will not sell goalkeeper David De Gea unless Sergio Ramos comes the other way.