Maguire, who is also attracting attention from Manchester City, delivered the transfer news during their pre-season training camp at Evian-les-Bains, according to the Daily Mail.
Leicester could demand as much as £90m for the England international but Manchester United are hoping a £75m compromise fee can be agreed.
Ole Gunnar Solskjaer is on the hunt for a central defender to partner Victor Lindelof at the back next term.
He is unlikely to face competition from Pep Guardiola for Maguire because the Premier League champions are unwilling to fork out £75m.
Maguire has established himself as one of the Premier League’s leading defenders after two seasons at Leicester.
He was part of the Foxes’ 31-man squad for the trip to France for warm-weather training.
Despite United failing to qualify for the Champions League, the Daily Mail claim that has not put Maguire off a move to Old Trafford.
He has reportedly stepped up his attempts to secure the transfer after Jose Mourinho’s failed pursuit 12 months ago.
United boss Solskjaer has admitted the club remain active on the transfer front after concluding deals for Daniel James and Aaron Wan-Bissaka.
“We are still in the market, we want more deals,” he said.
“We don’t have to sell, but we also don’t have to over-pay.
“Looking at our squad now, we have a good squad.
“This squad will have to be more robust once the season gets going.
“Last season we had a run without defeats, but the end was disappointing.”
Leicester are likely to hand Maguire the cash to sign a replacement for Maguire.
Brighton ace Lewis Dunk and James Tarkowski of Burnley are potential targets.
Foxes boss Rodgers already has Jonny Evans, Wes Morgan, Caglar Soyuncu and Filip Benkovic at his disposal.