The 22-year-old attacker has had an impressive season so far and made his debut for England in a 3-0 win over Wales in October.
United's scouting department have been impressed with the Foxes ace, and have been watching him closely.
United manager Ole Gunnar Solskjaer is also a fan of Barnes, and has been keeping a close watch on his progress.
And United could be tempted into making a move, with Borussia Dortmund still refusing to lower their £100m-plus asking price for Sancho.
It is understood Leicester value Barnes at around £60m, but would refuse any approaches for one of their star men in January.