The Gunners haven't won the English top flight in nine years now, and are still waiting for their first trophy triumph since 2005.
However, having taken 29 points from a possible 33 during their undefeated run at the end of last term, Walcott believes that Arsenal have what it takes to compete for the domestic crown.
"The teams that finished above us have changed their managers," Walcott told the club's official website. "We've still got ours. It's always going to be tough. Every game in the Premier League is very tough and very interesting. It always comes down to the last few games. So we need to believe in ourselves.
"The way we ended the season last year, 11 games undefeated - we need to do it from the start, which I believe we can. The manager shows so much belief in us in every single game so we need to step up and I think we've got a very good chance of challenging this season.
"I got 21 goals last season and I want to try and push on from there. I want to try and get to 25. That's my benchmark but I believe in myself and that's the main thing. Time will tell if players come in or not but we just work with the squad we've got at the moment. Come the start of the season, there could be changes. We know it's a very long campaign but we're all very excited about our prospects."
Arsenal host Aston Villa on the opening day of the Premier League season on August 17.