The Iceland international has made only two competitive appearances for Arsenal since signing from Djion this past summer.
"That should be the mindset for everyone, that you want to play," he told reporters.
"It doesn't matter what position you play. Maybe as a goalkeeper it's a little bit different but definitely you have to have the mentality to push to become No.1.
"If it happens in one week or one year that always has to be your mentality otherwise you're in the wrong profession I think."
Quizzed if he was capable of challenging Leno for a starting spot, Runarsson added: "Yes, 100 per cent. Otherwise I wouldn't be playing for this club.
"I have to have this mentality that I can play and I have to believe it."