With Andre Onana and Altay Bayindir both struggling since the start of the season, they're also losing the confidence of teammates ahead of them.
The Sun says United players believe Heaton is the best of three in training and want to see him called up by manager Ruben Amorim.
Bayindir has played all three of United's Premier League games this season and again was at fault during yesterday's win against Burnley.
Fit-again Onana, who missed the preseason with a hamstring injury, was called up for the Carabao Cup defeat to Grimsby and also made an error on the night.
A source said: “Judging it on performances on the training pitch, some of the lads think Tom deserves a chance.
"He’s very experienced, looks after himself and is a key member of the squad.
"The other two are both struggling at the moment, so why not give Tom a go?”