Arsenal, who last lifted the title in 2003/04 when they went through the league season unbeaten, lead the standings on goal difference from Leicester City but failed to pad their advantage on Sunday when they were held 0-0 at Stoke.
"The mental strength in this dressing room's the strongest it's ever been," Walcott told the club's website, adding that the arrival of goalkeeper Petr Cech from Chelsea last June had added some steel to the side.
"When you have the likes of Petr Cech come into the dressing room as well, he just enhances that. He's a leader in that dressing room as well.
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"We've just got to look at him because he's obviously been at the top level and won many things. We all have the belief he's got that we can push on."
With just seven points separating the leaders from fifth-placed Manchester United after 22 games, Walcott said the title race was wide open.
"You can't tell what's going to happen this year, it's just so tight," the England international said. "The amount of teams that could potentially win this league, it's probably going to be the most exciting Premier League for a neutral ever.
"No one can call it at this moment in time but we're top of the league and we can be proud of that."
Arsenal host champions Chelsea, who are 14th, in the league on Sunday.
(Reporting by Shravanth Vijayakumar in Bengaluru)
