The 22-year-old has had an impressive season so far and made his debut for England in a 3-0 win over Wales in October. And according to the Daily Star who quote the Sun in a far fetched rumour, United are considering making a move for the Englishman during January.
However, the report adds that Ole Gunnar Solskjaer's side will have to wait to bring Barnes to Old Trafford as Leicester seem reluctant to sell one of their top-performing players midway through the season.
Brendan Rodgers' side are in fourth place ahead of their trip to face Sheffield United on Sunday, and are targeting a Champions League spot after narrowly missing out to Chelsea last season.
Barnes has been in form for the Foxes this season, having scored three goals in the Premier League and two in the Europa League.
The report also claims that Leicester value Barnes at £60m, but would refuse to entertain any approaches for one of their most highly-rated players.
United still consider Borussia Dortmund's Sancho as their number one target, but the Bundesliga club are not expected to relent on their £108million price tag for the 20-year-old winger.