The 24-year-old, who almost joined Sunderland in the summer, has hardly featured for Sir Alex Ferguson's side this season.
Gibson was recalled for the Boxing Day victory over Wigan after United were hit by an injury crisis.
But his impressive display was overshadowed by a night out with team-mates Wayne Rooney and Jonny Evans.
Rooney was fined more than £200,000 and dropped for 3-2 defeat by Blackburn because the striker was thought to be the worse for wear and unable to train properly.
Now Everton boss David Moyes could make a bid for Gibson in the January transfer window.