The Irish international is on the verge of joining Sunderland on a permanent deal after a four and a half year spell at Goodison Park.
The 29 year old has yet to play a league game under Ronald Koeman this season.
It's believed Moyes is eager to sign Gibson as well as Toffees teammate Bryan Oviedo
If both deals go through, six of Moyes's former players at Everton will be playing for him at Sunderland.
Meanwhile Sunderland left-back Patrick van Aanholt has joined relegation rivals Crystal Palace.
He's agreed a four-and-a-half-year contract at Selhurst Park in a move that's believed to cost 14 million pounds.
