The 27-year-old, who left Brentford to return to his former side, admitted he felt there was "unfinished business" at The Hawthorns.
Celtic were understood to be keen on the Nevisian international with the Bees standing firm on a £3 million fee for the player.
The Daily Record reports that the champions did submit offers for the playmaker, which were rejected.
Speaking to West Brom's website, Sawyers said: “When the opportunity came up to return, I had to take it.
“It’s massive for me to be back. There is no other team that means as much to me as West Bromwich Albion. Everyone who knows me, knows that.
“I think this is unfinished business for both me and the club.”
Neil Lennon is understood to be targeting reinforcements in midfield, and has been linked with a string of players this summer including Adam Maher, Rekeem Harper, Jose Cifuentes and Sergio Quintero.