The Baggies have continued to look to add to their options despite the closure of the transfer window as they are still able to sign free agents.
Bryan is a player the club have zeroed in on and he passed a medical earlier today to edge towards sealing the move.
He is now set to sign a contract with the Baggies and will be at the disposal of boss Valerien Ismael.
Bryan, 24, was on the books at Sheffield United, but departed the club as a free agent before the transfer window closed.
As such he is free to make the move to the Hawthorns outside the transfer window system.
The former Manchester City youth player was signed by Sheffield United on a free transfer in the summer of 2018 and made appearances for the Blades in the Premier League last season.
Bryan, who can also operate in midfield, will now focus on trying to help West Brom bounce back up to the Premier League.
