Bartley made only 10 appearances for the Swans last season. He had to undergo a surgery in September after suffering a medial ligament damage.
The Baggies are desperate for defensive reinforcement and are keen to sign the £4million-rated centre-back. Jonny Evans already left the club, and with the futures of Craig Dawson and Ahmed Hegazi in serious doubt, the Baggies are looking at new faces to rebuild the squad.
The news will come as a blow to Leeds United who were heavily interested in signing him.
Back in April, the Yorkshire Evening Post reported that Leeds United are in negotiation with Swansea over the signing of Bartley.
He is a player that the large majority of the Leeds supporters would love to see back at Elland Road where he enjoyed a superb loan spell. However, as it stands, he is close to joining West Brom instead.