The 25-year-old has impressed for the Foxes since his switch from Hull City last summer, making 38 appearances in all competitions for the 2015-16 Premier League champions.
Jose Mourinho, meanwhile, is thought to be eyeing an overhaul of his defence at the end of the season, with the likes of Chris Smalling, Luke Shaw, Phil Jones, Matteo Darmian and Daley Blind all believed to be on borrowed time at Old Trafford.
According to The Mirror, Mourinho has been impressed with a raft of scouting reports on Maguire, who scored a last-gasp equaliser in December to deny United an away win over Leicester.
The report goes on to claim that the Foxes, who paid Hull £12.3m for the player's services last summer, could be persuaded to do business starting at £35m.
Maguire has also impressed for England in recent months and is expected to be in Gareth Southgate's squad for the Russia World Cup.