The Gunners are likely to finish third but are currently 13 points behind leaders Chelsea after Sunday’s 1-1 draw with Manchester United.
“Third place is a step forward from where we finished last year,” admitted Walcott.
“But we want to – and need to – be pushing on now and trying to challenge for the title.
“There was an opportunity in the season when we were getting close to Chelsea and people were talking about us competing but I sensed throughout the dressing room that we were never quite up there with Chelsea.
“But looking at the squad, the manager has an absolute headache to try and pick his best team and we want that going into next season.
“We are looking very strong and you never know, he might add a few extra players but we need to be pushing for the Premier League now. There is no excuse.”
Walcott admits that being largely restricted to the bench since returning from injury has tested his patience.
“Of course (it has been frustrating),” Walcott said.
“Any player wants to play. I want to play. I have been out for a long time. The manager was patient, I have been patient myself.
“I haven’t expected to be where I was before I got injured. But when given the opportunity, I need to be ready and focused at all times because it is not just about the starting line-up, it is about the whole squad.”