Sancho was United’s top transfer target in the summer but Borussia Dortmund stood firm over their €120 million (£108m) of the England international.
United were unable to agree a deal with Dortmund and ended up signing two inexperienced wingers in Facundo Pellistri, who is yet to make his debut, and Amad Diallo, who has remained with Atalanta until January.
According to The Sun, United have been monitoring Barnes’ progress and have been impressed with the 22-year-old’s performances.
The report claims that Ole Gunnar Solskjaer is also a huge admirer of Barnes and has been keeping track of his displays for Leicester.
The Foxes reportedly value Barnes at £60 million, but are unwilling to let him go in the January transfer window.
Barnes, who came through Leicester’s academy, has scored three goals and registered one assist in 10 Premier League appearances this season.
The 22-year-old also earned his first senior England call-up in October and made his first appearance in the 3-0 win over Wales.
‘He’s spoken about the progression from stepping up from being a young player to becoming a top player,’ said Barnes.
‘He’s reiterated to me that if you are an attacking player in the final third, you can play as well as you want but at the end of the day you need to be getting the goals, the assists, the chances created and hitting those levels.
‘Otherwise you’re not helping the team in the right way.
‘You see the top wingers score goals, it’s one of the main things that they score a lot of goals. So I want to base my game on them and try to hit the levels that they’re hitting. It’s obviously a big challenge.’