Barca say Bayern will pay 8.5 million euros (£7.8m) and Coutinho's wages, and that the option for the Bundesliga champions to then buy is fixed at 120m euros (£109.8m).
The 27-year-old Brazil playmaker passed his medical with Bayern on Sunday before putting pen to paper on his loan contract, and following the deal being confirmed, was then unveiled at a press conference on Monday afternoon.
He has been given the number 10 shirt previously worn by Arjen Robben, who retired during the summer.
And Coutinho was quoted on his new club's official website as saying at Monday's press conference: "The jersey of FC Bayern and the number 10 is a big responsibility.
"It belonged to Robben, he was a great reference in football. I hope to be able to confirm expectations with good football on the pitch."
Coutinho joined Barca from Liverpool in January 2018 in a £142m deal but has struggled to reproduce the kind of form he showed for the Reds.
He has scored 21 goals in 75 games in all competitions for Barcelona and produced 11 assists, winning LaLiga twice and the Copa del Rey.
Regarding Coutinho's number at Bayern, chief executive Karl-Heinz Rummenigge said: "We planned to not award the 10 and seven this year as a gesture for Robben and (Franck) Ribery (who also left the club over the summer).
"We had a conversation with Robben and we wanted to give Philippe his number as a token of appreciation, and Hasan (Salihamidzic, Bayern's sporting director) showed him the jersey at the first meeting.
"Arjen had no problems with that, he said Philippe was a worthy successor."
Coutinho said the '10' role was his "favourite position" before stressing that "the coach (Nico Kovac) decides where to play"
He added: "I want to be ready to play as soon as possible and I cannot wait for tomorrow with the first meeting with my colleagues and the coaches.
"There are always big expectations at Bayern and of course the Champions League is a goal and I will do my best to win the Bundesliga, the cup (the DFB-Pokal) and the Champions League with my colleagues.
"I have a big challenge here and I hope to stay here for a long time and win many titles.
"I was presented with an ambitious project, I'm an ambitious guy, and I talked to Thiago, Rafinha and Lucio, and I've heard a lot of good things, they really helped me a lot.
"I'm hardworking and ambitious, and every time I wear the Bayern jersey, I'll do the best I can on the field to reach our goals."
Rummenigge said: "We are convinced that we have brought a very good player and I do not rule out that it can be a long-standing partnership."
Salihamidzic added: "Philippe will strengthen our team and give our game that spectacular touch, which we all look forward to."
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