The 31-year-old was deemed surplus to requirements by Premier League champions Manchester City at the end of last season and they decided not to offer a new deal.
Following City's final game of last season, the player wrote on Twitter: "A very emotional time for me after the game."
Lescott had been a target for Stoke City and Hull City but is undergoing a medical at The Hawthorns ahead of a potential two-year contract.
The England international played for the Baggies’ fierce rivals Wolverhampton Wanderers between 2000 and 2006 before a move to Everton, where new Baggies boss Alan Irvine had a coaching role at the time.
Mark Hughes bought Lescott for City for a fee of £22m in August 2009; he has won the FA Cup, the League Cup and two Premier League titles at Eastlands.
“There is an interest in Joleon and he is a player I know well,” Irvine told a media conference on Wednesday.
“I obviously worked with him at Everton and I remember there were a number of times we watched him.
“Coincidentally, I saw him towards the end of the season when Manchester City played Everton. I had a chat with him then but, of course, I didn’t know any of this was happening at the time.”