The Gunners have struggled to challenge for the top spot in recent seasons but Walcott believes that Arsene Wenger's men should not be content with qualifying for the Champions League each season.
"For a club like this, with the stadium, the support that we have - we have to be that," Walcott told ESPN.
"We don't want to be [just] challenging for that fourth spot anymore. We want to challenge for the Premier League and that's our main goal.
"We know Arsenal's a massive club and where we need to be. What we need to do is get to the next level and compete a little bit more -- not just getting into the Champions League. The next step is to compete."
Arsenal face Tottenham Hotspur in the Premier League on Sunday afternoon.