Kaveh Solhekol of Sky Sports said United know it will require a "world-record bid for a defender for Leicester to be tempted to sell."
Maguire has risen to prominence courtesy of a strong season with the Foxes after joining from Hull City in June 2017.
His performances earned him a place in England's 2018 FIFA World Cup squad. He played a key role in Russia and scored his first international goal in the 2-0 quarter-final win over Sweden.
Maguire is a powerful, commanding centre-back who has emerged as one of the best defenders in the Premier League. However, he does lack top-level experience and has not played UEFA Champions League football.
The Red Devils are not short of defensive options, but manager Jose Mourinho has not settled on a favoured central partnership.
The club spent big on Victor Lindelof last summer, yet he has failed to become a regular in the team. Phil Jones, Chris Smalling and Marcos Rojo have all been guilty of making errors, while Eric Bailly found himself out of the side at the end of last season.
The club also remain linked with a move for Alderweireld, who may be a cheaper option, according to Marca's Chris Winterburn:
The Belgium international is an experienced operator, although he spent a large part of last season out injured and will celebrate his 30th birthday in March.
Journalist Tom McDermott offered his view on Manchester United's defensive targets:
Tom McDermott @MrTomMcDermott
Alderweireld my choice but from a business viewpoint I can understand if owners/EW are reluctant to spend a ridiculous amount on him because of age. Then there’s Maguire, who is a great presence in either penalty area/good with ball, but not sure 1v1. Wouldn’t bother with Mina. https://t.co/CtmfCKswgn
Leicester City manager Claude Puel said he is "confident" the club will keep hold of Maguire, per Shona Duthie at Sky Sports News.
Meanwhile, Manchester United have withdrawn their interest in bringing Mina to Old Trafford, according to Goal's Ignasi Oliva Gispert.
Mourinho has said his team face a tough campaign if they do not strengthen, as shown by Samuel Luckhurst of the Manchester Evening News:
A defensive addition looks likely before the close of the transfer window on Thursday. Alderweireld and Maguire both appear to be targets, but United will have to spend big to land either player.