Last week, three Adidas-backed clubs had offered $339 million for Barcelona's star player, but had all been rejected.The offer by the clubs was less than Messi's buyout clause with Barcelona, which stands at $345 million.
Bayern are allegedly planning to make a new offer this June, backed by Adidas, who have agreed to pay half of the costs of his deal to sweeten his possible move from Barcelona, says Paris-based Le10sport.com.
Additionally, two other clubs, one of them potentially being Chelsea, will also return with offers.
Messi is currently still in talks with Barcelona to upgrade his contract, a move which would make him the world's highest paid player, ahead of Real Madrid forward Cristiano Ronaldo.
The Argentinian's contract with Barcelona runs until 2018, when he will be 31.