The Spanish defender, who is being linked with PSG, has been in and out of the team over the past few years due to injury problems.
He has not been in the best of form this season, and Barnes thinks he is not the caliber of player the club should aspire to retain.
The 57-year-old told BonusCodeBets: "If [Belleri]) said he doesn't want to stay at Arsenal, then they should let him go.
"The team should only be interested in players who want to play through both good and bad times.
"Bellerin is part of the reason as to why Arsenal is a club in transition and not currently playing well.
"Players always aspire to play in the best team in the world and win loads of trophies, but if everybody did that then all the best players would be at one club.
"If your team isn't performing well, you want players who want to help the team perform better not ones who talk about leaving mid-season when the team is going through a rough patch.
"He owes more to Arsenal than to do that."