The 36-year-old will break Ryan Giggs's record of 632 Premier League appearances should he feature for the Baggies against Arsenal on Monday night.
The Englishman told Sky Sports News: "It's definitely a possibility. I was 32 when I signed for Everton and Roberto Martinez said, 'Your style of game - you can play until you're 40'.
"I'm sitting there laughing at him, but he was deadly serious, and I still laughed. It's still going to be tough, but for a manager to tell me that four to five years ago is a good compliment, which was nice to hear.
"I'm immensely proud to get there. Whether it will stay around for long I don't know."
Barry, who joined Albion from Everton in the summer, has a year's option on his initial 12-month deal.