Brendan Rodgers still believes Liverpool can gain a top four finish in the Premier League and earn Champions League qualification in order to retain the club's star players such as Luis Suarez.
The Reds are struggling in ninth place after being beaten 2-0 at home by West Brom on Monday, and with 36 points from 26 games they are well short of Tottenham, who have 48 points with the same number of matches played.
But in an attempt to inspire confidence in his side, the Anfield boss insists he will continue fighting to get the club back into Europe's premier club competition until it is a mathematical impossibility.
"For me, I will always say that something is possible if it's mathematically possible," he told reporters. "Say, for example, we won 10 of our last 12 games then 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 is going to be very difficult. I don't want to be hung out to dry as a fool.
"You can live without water for so many days, but you can't live a second without hope. If you never have hope and want to achieve something, then you've got no chance. While we can mathematically do it, I retain that optimism."
Rodgers believes that being optimistic despite the pressures of trying to meet the high expectations of Liverpool fans will in turn help the players to overcome their doubts, and thus improve performances on the pitch.
"The reality is, it's still there because the points mean it is still achievable," he continued. "I have got to where I am in life because I am optimistic. For me, the glass is always half-full.
"I have tried to promote that way [of thinking] to the players because if you don't think you can do it, you'll never do it. It is as simple as that."
However, the Northern Irishman also admits the prospect of another season without Champions League football may lead to the club losing some of their more established names, adding: "I am a realist."
Liverpool take on Swansea at Anfield on Sunday; a win for Rodgers' side would take them above Michael Laudrup's Swans into seventh, but a defeat or a draw would all but end any realistic hopes of a push for fourth.