United moved ahead of rival Manchester City in the chase for the England defender by increasing its offer from the initial £65 million that both clubs had indicated they would be happy to pay a fortnight ago, Sky Sports News reported.
But Leicester has made it clear it is not interested in selling at that price, especially after seeing United pay £50 million for Aaron Wan-Bissaka, who The Foxes feel cannot compare to Maguire in proven international quality and experience.
Aiyawatt Srivaddhanaprabha, Leicester's owner, would need to personally sign off any deal for Maguire, who still has four years left on his contract and is seen as a key asset in Leicester’s squad.
Maguire – along with all of the club's other international players – was not in the group who returned to begin Leicester's preseason training schedule on Monday at their Belvoir Drive training complex.
It is not yet known if the England defender will join his teammates on Sunday when they head to Évian-les-Bains for six days of training in the French Alps.
United's chances of landing Maguire may be influenced by City defender Nicolas Otamendi, who was expected to leave the Etihad this summer for Atletico Madrid. However, he may reconsider his decision after the departure of Vincent Kompany.
Maguire is understood to be happy at Leicester and will not attempt to force a transfer, but is open to the idea of a move to one of the Premier League's leading six sides.
Leicester's position has always been that it has no desire or need to sell and would only consider a bid in excess of the £75 million world-record fee paid by Liverpool for Virgil van Dijk.
However, it will welcome the prospect of two big clubs bidding against each other for one of their prized assets.