The 36-year-old is expected to start the Baggies' Premier League meeting with the Gunners on Monday evening, marking his record 633rd Premier League game.
Ahead of the milestone occasion, the former England international has revealed that he came close to joining Arsenal when leaving Aston Villa for Manchester City in 2009, and again five years later when he headed to Everton.
"There were a couple of conversations with representatives and Arsene that he was interested," he told reporters. "Before I was leaving Villa and when I was leaving Man City to go to Everton there was a bit of interest too. But it didn't materialise.
"I've got massive respect for Arsene Wenger for what he's done for the English game and it would have been good to work under him, but that didn't happen.
"He deserves credit because he was ahead of the game back then. If you speak to people who worked with him in the late 1990s, he was seeing things and doing things that other managers in England weren't."
Barry featured 365 times for Villa in the Premier League, on 132 occasions for Man City and 131 times for Everton, who he left to join West Brom in the summer.