According to Kicker, the deal is nearing completion after a lengthy negotiation process, with the 29-year-old previously tied to the club only until 2027.
Talks had dragged on for months due to a gap between the two sides over salary expectations.
Laimer is said to have initially sought a rise to between €12m and €15m a year, with the Austrian currently reported to earn between €8m and €9m.
A more modest increase now appears to have been agreed.
According to Kicker, player and club have largely settled on terms that are economically sound for Bayern Munich following productive talks, with official confirmation expected in the near future.
Laimer has established himself as a regular starter at full-back under head coach Vincent Kompany, having joined Bayern on a free transfer from RB Leipzig in 2023.
He has made 136 appearances for the club since arriving, valued for his versatility across multiple defensive and midfield positions.
Bayern had previously shown some reluctance to meet Laimer's initial demands, with honorary president Uli Hoeness addressing the situation publicly during the recent run-in.
"Konni is a player I value highly. He's extremely important for the team, as well as for the club's public image. He works incredibly hard for the team, but he isn't Maradona. And players like that have to accept that there are limits."
Sporting director Max Eberl had also acknowledged the gap between the two parties during the closing stages of the Bundesliga season.
"There are two positions that currently don't align. We want to extend with Konni, that's a fact. Now we somehow need to find a bridge in terms of expectations."
Should the agreement be finalised, Bayern will have secured their full-back position with greater certainty for the coming seasons, with Josip Stanisic also under consideration for the role.