Wenger's contract at the Emirates Stadium expires at the end of the season and some Gunners supporters feel that a change is needed after nine years without silverware.
They may yet end that drought with victory in the FA Cup later this month, and Gilberto, who played for Arsenal between 2002 and 2008 and was a member of the Premier League-winning 'Invincible' team, believes that would ease some of the pressure on the Frenchman.
Regardless of the result against Hull City at Wembley, though, Gilberto feels that the seven major trophies that Wenger won prior to 2005 should earn him more patience from the fans.
"It is not an easy job because there is a lot of pressure with Arsenal not winning the title for a few years but this season they have an opportunity to win The FA Cup," he told BT Sport's SportsHUB. "I really hope they can do it and this will then of course ease the pressure.
"This kind of pressure is normal when you have been as successful as him in the recent past for the club. Some people say he should leave but I disagree because he brought success to the club and people shouldn't only analyse the last five-eight years but also look at what he has achieved throughout his whole time at the club."
Wenger has said that Arsenal "have his word" over signing a new contract to extend his 18-year tenure.