The centre-back, 25, played a key role in England’s progression to the World Cup semi-finals this summer, starting every game in the knockout stage.
He played every Premier League minute for the Foxes last season after a £17m move from Championship side Hull City.
United manager Jose Mourinho is keen to sign a central defender and is a known admirer of Maguire after his standout performances in Russia.
And with the transfer deadline on 9 August this season, the Foxes are privately making their position on Maguire clear.
United have so far signed three players this summer – Brazil midfielder Fred for £47m, Portuguese defender Diogo Dalot and third-choice goalkeeper Lee Grant.