The 27-year-old has been linked with a move away from the Allianz Arena during the current transfer window, with former club Barcelona among the teams rumoured to be interested in his signature.
However, ESPN reports that the strongest interest has come from the Premier League, with United and Chelsea both considering an official approach prior to the August 9 transfer deadline.
Bayern are reluctant to sell the Spain international, but the player himself is thought to be open to the idea of ending his five-year stint in Munich.
Bayern are also looking to offload one or two midfielders this summer, as they currently have Thiago, Javi Martinez, Arturo Vidal, Sebastian Rudy, Corentin Tolisso, Leon Goretzka, James Rodriguez, Thomas Muller and Renato Sanches as options in that area of the pitch.
However, the German champions would demand at least €70m (£62.2m) before considering a bid for Thiago, who has helped the club to five successive Bundesliga titles.
Both United and Chelsea are also understood to be interested in Vidal, who has been left out of the club's squad for their pre-season tour of the United States. body check tags ::