The 29-year-old informed the Bundesliga giants before the end of last season that he would not be renewing his deal at the Allianz Arena, which runs until 2021.
Liverpool had been favourites to sign Thiago, with Jurgen Klopp holding talks with the Spaniard earlier this month.
However, Klopp must sell a senior player before Liverpool’s board sanction a £27million move for Thiago but the German is content with his squad.
Gini Wijnaldum appears to be the only player that could possibly leave Anfield this summer but Klopp wants to keep the former Newcastle midfielder.
Liverpool’s reluctance has opened the door to Manchester United, who have long coveted Thiago.
According to Bild, United have not yet made a bid for the former Barcelona midfielder but they have contacted his representatives to discuss a move to the Theatre of Dreams.
Thiago merely wants to experience a new challenge and though he would like to work with Jurgen Klopp, he is open to a move to rivals Manchester United.
The Germans want £27m for the former Barcelona midfielder, who is currently on international duty with Spain.
United kicked off their summer spending with the £40m capture of Ajax ace Donny van de Beek this week but Ole Gunnar Solskjaer remains keen on adding further reinforcements in midfield.