Two defeats in their last three league outings have seen City fall six points behind their rivals, having played a game more.
Many fancy Chelsea to cruise to a fourth Premier League crown, but Hart says City will continue to fight right to the end.
"I will never give up, ever, and I could safely say that nobody in our dressing room will ever give up either. We will see where that takes us," he said.
"West Brom is a huge game and we turn our focus to that. We have no excuse not to throw everything at the Premier League now and give our all - to the fans, the owner, to everyone.
"Every game is a must-win now. We have no time to lick our wounds, we have to fly into that game.
"We have to win every game, and that is how we will go into it, and see where it takes us.
"We have had a few knocks this season, but we have proved it time and time again, and so have our fans, that we come back and we keep fighting."
