The Austria international midfielder will sign a four-year deal with the Bundesliga champions when his contract with Leipzig expires on July 1.
Laimer joined Leipzig from RB Salzburg in the summer of 2017 and made 190 appearances, scoring 15 goals and winning the DFB-Pokal on two occasions.
The 26-year-old will be Thomas Tuchel's first signing as Bayern Munich boss since replacing Julian Nagelsmann in March.
"It's a dream come true for me," Laimer said after completing his move.
"FC Bayern is one of the biggest clubs in the world.
"I will give everything for the club and the fans. FC Bayern always has the highest goals - just like me. I'm in the right place here. I'm only capable of giving 100%, and that's what I'll put into every training session from day one. I can hardly wait."
Bayern CEO Jan-Christian Dreesen added: "We are delighted to have signed Konrad Laimer. He comes to us as a two-time DFB Cup winner - and together we will now hopefully celebrate many more great victories and titles here at FC Bayern."
Laimer has won 26 caps for the Austria national team.
