Bayern sign French winger Olise from Crystal Palace
German magazine Kicker reported Bayern paid a release clause of 70 million euros (£59.3m) to the Premier League club to secure Olise's services on a five-year deal.
Olise, 22, completed a medical in the Bavarian capital on Sunday.
"It's a great challenge, and that's exactly what I was looking for," Olise said.
"I want to prove myself at this level and play my part in ensuring that we win as many titles as possible with our team in the coming years."
Born in London, Olise has represented France's youth teams and will play for Les Bleus under coach Thierry Henry at the Paris Olympics in August.
Olise scored 10 goals and provided six assists in 19 Premier League games last season for the London club.
He is Bayern's second major arrival of the summer after Japan defender Hiroki Ito who arrived from Stuttgart.
Bayern finished third behind champions Bayer Leverkusen and Stuttgart last season, breaking a streak of 11 straight Bundesliga titles.
Other than England captain Harry Kane, who scored 36 goals in 32 Bundesliga games, Bayern struggled for goals last season, with only Jamal Musiala (10 goals) reaching double figures.
"We want new impetus in our team, new energy, new ideas – that's what players like Michael Olise stand for," Bayern sporting director Max Eberl said in a statement.
"We're very much looking forward to having him here. He will enrich our game."
Bayern are also reportedly close to finalising a deal for Fulham midfielder Joao Palhinha, while Leverkusen defender Jonathan Tah and RB Leipzig forward Xavi Simons are also on their shopping list.
