Alcantara, 29, has won the Bundesliga every season with Bayern, making 230 appearances for the German giants who were last week confirmed as 2019/20 champions.
He joined from Barcelona in 2013, when Bayern activated a release clause in his contract for 25 million euros ($28 million).
According to both Sport Bild and Sport1, Bayern offered him a contract extension in early May, which Alcantara is reluctant to sign as he wants to join another big club.
Sport Bild says Liverpool is a likely destination as the Reds’ manager Jurgen Klopp is known to be a fan of the technically-gifted Spaniard.
Surgery on a groin injury has sidelined Alcantara and in his absence, Leon Goretzka and Joshua Kimmich have dominated Bayern’s central midfield berths.
However, Bayern will not want to let go of Thiago, who is one of their key playmakers in attack and often appears in the right place in defence.
“He is an exceptional talent. I would like him to stay,” said Bayern head coach Hansi Flick recently. .
Thiago returned to training on Monday for the first time since his operation and Flick hopes he could still feature this season.
“We are hoping he will be fit soon, maybe even for the cup final,” Flick added with Bayern to face Leverkusen on July 4 in Berlin for the German Cup.