Bergkamp, who will complete 10 years with the Premier League champions when his current one-year deal expires at the end of this season, indicated earlier this week he might seek to play on to 2006.
Wenger told reporters on Friday: "For Bergkamp, the case is quite simple. We always sit together in April or May and we have a frank discussion -- and that will happen."
Pressed on whether he would offer a new deal to the Dutchman, who joined in 1995 after a two-year spell at Inter Milan, the Arsenal boss said: "I would not rule that out."
Wenger also said he was hoping to agree new contracts with England central defender Sol Campbell and Brazilian midfielder Edu.
Campbell, 30, signed a four-year deal in 2001 when he joined from north London rivals Tottenham Hotspur on a free transfer and his representatives have been in talks with the club in recent months.
"We are closer," Wenger said. "If you are going from Newcastle to London, we are at Watford.
"Sol, I feel, will make some sacrifices to stay with us. But we can compensate by giving him a longer period of contract."
Edu, who has been linked in media reports with a January move to Barcelona, is also expected to sign a new deal although Wenger admitted the absence of a signature to date was a worry.
"We need his commitment," Wenger said of the 26-year-old. But he added: "My feeling is that he wants to stay."