Jose Mourinho is keen to land a new centre-half this summer.
The Red Devils have been tipped to sign Tottenham star Toby Alderweireld, however, talks with Spurs are at a standstill.
Tottenham are demanding around £60m for the Belgium international, 29, this summer.
Reports this week claimed Mourinho has turned his attention to England’s World Cup hero Maguire as an alternative to Alderweireld.
And the bookies have slashed the odds on Maguire quitting Leicester for Old Trafford from 10/1 to 11/10.
Betway’s Alan Alger said: “Harry Maguire was one of the shining lights for England at the World Cup and his performances have clearly caught the eye of Jose Mourinho and the Manchester United hierarchy.
“Mourinho’s side are now just 11/10 to sign the Leicester defender, having been available at 10/1 a few days ago.
“Maguire is still 8/11 to remain with the Foxes as Claude Puel plots improving on his side’s ninth place finish last season.
“While a move to Tottenham looks unlikely with Spurs out to 13/2 having been the 2/1 favourites to sign him.”
Should United land Maguire, he will become Mourinho’s fourth summer signing.
The 25-year-old would follow Fred, Diogo Dalot and Lee Grant to Old Trafford.