The 27-year-old Croatian international has been linked with a move since Bayern announced the signing of Robert Lewandowski.
Lewandowski, a one-time Manchester United target himself, will join the defending Bundesliga champions at the end of the season.
And reports have suggested Mandzukic could be deemed surplus to requirements by Bayern boss Pep Guardiola this summer.
Sky Bet have made United 4/1 favourites to sign the Croatian star before Friday's deadline, while the next best priced team are Arsenal at 7/1. Chelsea are 10/1.
He was named on the bench for the trip to Stuttgart last night, prompting speculation that Guardiola is ushering him towards the exit door.
There had been rumours that the former-Barcelona manager was unhappy with Mandzukic's application in training, and he's no longer the first-choice starter he was under Ottmar Hitzfeld.
United made a big splash in the transfer market earlier this week when they confirmed the signing of Juan Mata from Chelsea.
The £37.5m deal is a club record for United, and the reigning Premier League champions carried the momentum into Tuesday night's 2-0 victory over Cardiff City, during which Mata made his debut.