A run of just 12 points in 14 games has seen the Premier League champions slump to eighth place in the table and left their title defence in tatters.
"In our heads we always believe 'never give up' - and we won't give up," said the Senegal international.
"The human being in us will never give up. As a football player 'never give up' should be your first expression. It is my expression.
"Our target was to win the league. At the same time, if you are not winning and Manchester City is flying to win every single game, to win it is very complicated. So we will try to do everything possible to be in the top four."
Mane added: "As a professional footballer it is not easy (to have a bad run) when you are used to winning every single game like we were used to last year.
"At times it was tough, but not like it is now, which is a really complicated time.
"But we are more experienced now and we know how to deal with things. You just have to stick together and be as positive as you can."
Lifting the Champions League trophy for the second time in three years is another route back into the competition, and the forward said: "Being in the top four right now is our main target, but to win the Champions League is our main, main target.
"It won't be easy because we are going to face the best teams in Europe, but we are here to beat the best in the world.
"We will do everything possible to go for it and to win the Champions League."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!