The Reds are currently ninth, 12 points off a Champions League qualification place, with just 12 games to go.
They lost 2-0 at Anfield to West Bromwich Albion in their last match, but Rodgers claims that while it is still mathematically possible, he will have faith.
"For me, I will always say something is possible if it is mathematically possible," said Rodgers. "Say, for example, we won 10 out of our last 12 games, that would take us close to fourth place.
"While there's a chance I have to retain my optimism.
"So while we haven't given up, we also know it's going to be very difficult. I don't want to be hung out to dry as a fool."
Liverpool face Swansea City in the Premier League this afternoon.