The 26-year-old rewarded his manager’s decision to start him ahead of Olivier Giroud at Wembley by scoring the opening goal as Arsenal powered to a comprehensive 4-0 victory over Aston Villa.
Walcott has been linked with Liverpool and Manchester United after failing to sign an improved deal during an injury-plagued season.
But after scoring a hat-trick against West Brom on the final day of the Premier League season and lifting the FA Cup, Wenger insists he plans to keep hold of the England international.
“I don’t see why he should not be here next season,” Wenger said.
“He is a player who was out for a year – January to January – and it took him a while to come back to his best.
“I think he is now at an age where it is very interesting. He now has experience, he’s at the peak physically and it’s time to stay with us.
“I had many, many big decisions to make before the game today. I had to leave big players out of the squad. I had to make difficult decisions to try to find a good balance in the team.
“Walcott was on a high because he scored a hat-trick last week and has come back in good form. Giroud has given a lot to the team since January, but in recent games I think he was a bit jaded and tired.”
Wenger says his players can use a second successive FA Cup victory as a springboard to win more trophies next season.
But speaking minutes after lifting the trophy at Wembley, the Frenchman was unwilling to discuss any potential summer transfer targets.
He added: “I’m not in a buying mood tonight. I will buy a drink for my players!
“We will see what’s going on in the transfer market. We have a good basis and we have some time now to prepare because we don’t play a qualifier in the Champions League.
“We have a good confidence level. Let’s show we can be intelligent during the summer.
“Of course we can push on. Why not? We have won the league in the past when we had the potential to do it and that’s what we want to do again.”