New United boss Erik Ten Hag may yet decide that Garner should be part of his Old Trafford revolution.
But Leicester and Everton are the most likely clubs to pounce if the England Under-21 international is allowed to leave the club he joined when he was eight.
Garner, 21, has attracted a number of admirers with his performances on loan for Nottingham Forest.
Southampton and Leeds have also been keeping tabs on him.
But Foxes boss Brendan Rodgers has made it clear that he wants to reinforce his midfield and is believed to have Garner high on his list of potential signings.