According to German media reports, Bayern will pay 60 million euros, including add-ons, to bring the London-born player to the Allianz Arena.
Olise scored 10 goals and provided six assists in 19 appearances last season, despite a campaign impacted by injuries.
"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," Olise said.
He was reportedly pursued by several clubs, including Chelsea and Newcastle United, before ultimately choosing Vincent Kompany's side.
Despite being born in England, Olise has played seven matches for the France Under-21 side, having a French-Algerian mother. He has been selected for the France squad for the Paris Olympics, which start in July.
