While Barcelona are the United No. 10's dream destination, the La Liga giants are prioritising a swoop for Athletic Club winger Nico Williams.
And Bayern reportedly could take advantage of Barca's focus lying elsewhere as they weigh up a push for Rashford.
Reports in Germany claim the Bundesliga champions have emerged as potential suitors for the United outcast, who doesn't feature in Amorim's future plans.
Villa chose not to exercise an option to sign Rashford for £40million at the end of his loan, confirming the decision on Monday.
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It has left the 27-year-old facing an uncomfortable return to United when pre-season begins at Carrington on Monday.
Villa's decision and Barca's priorities have paved the way for other clubs to swoop for Rashford, with Bayern manager Vincent Kompany contemplating a move for the England international.
Bayern are believed to have been impressed by Rashford's resurgence in form at Villa, where he netted four goals and provided six assists before his season was cut short due to injury.
Kompany is eager to strengthen his attacking line-up this summer, and Rashford is on his radar.
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The United academy graduate could be set to team up with England team-mate Harry Kane at the Allianz Arena.
Rashford's hefty weekly wages of £300,000 are the main hurdle to him securing a permanent move away from United this summer.
He will likely need to accept a wage reduction to orchestrate a departure to Bayern or elsewhere.
If Rashford is reluctant to accept a wage reduction or if clubs can't meet his current salary, United may have no option but to seek another loan deal next season.