The teenage winger, who lit up last year's Under-17 World Cup, submitted a transfer request on Wednesday in order to push through his switch to the Allianz Arena
The Bavarian giants have paid €500,000 for the versatile 18-year-old, who can play on either flank, in midfield or at full-back, according to Bild.
Although Weiser has been linked with Bayer Leverkusen in recent months, the player has publicly expressed his preference to join Bayern.
It was initially thought that Champions League runners-up might be willing to wait until next summer in order to sign the Germany Under-18 international for free, but Weiser submitted a transfer request on Wednesday and his move to the Allianz Arena look set to be confirmed in the coming days.
The teenager, who is a product of Koln's youth system, first came to prominence with a string of scintillating displays for his country at last year's Under-17 World Cup.