Adomah has netted five times in 20 appearances for the Bees, after joining from non-league side Harrow Borough at the start of the year.
Two Championship clubs have made offers for the player according to Barnet, but they have not named his suitors. They also claim to have received 'several enquiries' for his services.
The Scunthorpe Telegraph claim that the Iron manager Nigel Adkins is a 'known admirer' of the player, but do not say that United are one of the two sides to have made their interest formal.
Paul Fairclough, the Barnet manager, thinks Adomah has a bright career ahead of him in the game.
He said: "This player's future lies in the Premier League. He has got a future and a half.
"Clubs are already looking at him. He has so much belief and I'm surprised how quickly he has settled in at this level."