Tottenham are three points outside the UEFA Champions League qualifying places following a 2-1 defeat at Crystal Palace last weekend.
That loss at Selhurst Park was the first for Mauricio Pochettino's side since early December – an unbeaten run of six Premier League games having included a 5-3 demolition of leaders Chelsea.
And Walker says the performance against Jose Mourinho's men shows they have the quality to finish above the likes of Arsenal and Manchester United in the race for a UEFA Champions League place.
"Arsenal are always going to be up there as they have fantastic players but so have we," he told the Evening Standard. "We've got a dressing room full of internationals and good-quality players, who have shown this season what they can do.
"[The] Champions League is where every player wants to be but the way we're going now, I think we can get there.
"I don't want to jinx anything but if we play as we did when we beat Chelsea 5-3, why can't we do it?
"When Chelsea came to White Hart Lane, we proved that it wouldn't be an easy game for them. It will be a battle for the top teams. If they want to beat us, they're going to have to out-work us, out-pass us and out-play us.
"In recent seasons, we'd struggle to pick up points against the top teams and that's where we were struggling to bridge the gap but this season we've beaten Chelsea, drawn with Manchester United and Arsenal.
“If we can keep getting over that bridge, the gap to the top sides will be smaller and we will be right up there. When you look at how young the team are, I think the future is very bright."
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