The Republic of Ireland international has fallen down Sir Alex Ferguson's pecking order at Old Trafford, having made just two appearances for the club this season.
Toffees manager David Moyes only has a meagre budget to work with during the transfer window and is also thought to be looking to sign a striker and a central defender following injuries to Phil Jagielka and Sylvain Distin.
Moyes is yet to bolster his squad so far this month, with a deal to bring USA international Landon Donovan back to the club on loan from LA Galaxy having been set up before the window opened.
Gibson could be his first signing after seeing a proposed move to Sunderland fall through over the summer.
The 24-year-old came through the youth ranks at United and has previously spent time on loan at Royal Antwerp and Wolves.