The 36-year-old will play in his 633rd top-flight game and eclipse Ryan Giggs' record if he features against the Gunners on Monday.
Arsenal boss Arsene Wenger admitted he wanted to take Barry to north London when he left Aston Villa to join Manchester City in 2009 but could not compete with City's financial muscle.
The Gunners were also keen before Barry joined Everton five years later and the player acknowledged it would have been good to work with Wenger.
"There were a couple of conversations with representatives and Arsene that he was interested," Barry said.
"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 materialize. 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 made 365 top-flight appearances for Villa, 132 for Manchester City, 131 for Everton and has played four games for West Brom since signing for around £800,000 in the summer.