A 3-1 weekend defeat to Leicester City, after Mohamed Salah gave the Reds a second-half lead at the King Power Stadium, led Klopp to concede his defending champions will be unable to catch Premier League leaders Manchester City.
In their previous match, Liverpool were beaten 4-1 at home by City, who are 13 points better off with a game in hand.
Attention now turns towards the Champions League, where Liverpool will look to continue an impressive recent record in Europe's premier competition when they face last season's semi-finalists RB Leipzig.
The rumour mill went into overdrive after the Leicester defeat, with unsubstantiated online speculation suggesting Klopp might step down amid dressing room ructions.
The 53-year-old also experienced personal tragedy when his mother Elisabeth died last week, with coronavirus travel restrictions preventing him from attending the funeral.
Despite being kept away from matches, a group of Liverpool fans looked to rally around Klopp by hanging a banner in his honour outside the Kop end of Anfield - a gesture he appreciated but felt was unnecessary.
"The banner is nice, not necessary I don't think. I don't feel like I need special support in the moment [but] it's nice, very nice," he said ahead of Tuesday's match, which has been moved to Budapest.
"What was the other thing? That I get the sack or leave by myself or what? Neither, nor. I don't need a break.
"The last thing I want to do is talk about a private thing in a press conference. Everyone knows that privately we [Klopp's family] had a tough time. That wasn't only for three weeks, that was a much longer time.
"We always deal with it as a family 100 per cent.
"When I come here at the football club to work, I can split things. I can switch off. I don't carry things around. If I'm private, I'm private. If I'm in my work place, I'm here. Nobody has to worry about me.
"I might not look like this because the weather's not cool and the beard gets more and more grey, I don't sleep a lot. That's all true. But I'm full of energy.
"The situation is a challenge. I don't want the situation but, now we are in it, I see it as an interesting challenge."
Liverpool's five Premier League losses since the turn of the year is one more than they suffered over the course of the previous two completed seasons.
Klopp is realistic over the magnitude of such a fall but insisted he and his players are ready to roll up their sleeves.
"Nobody wrote a book about being in a situation like this and how you sort it," he said. "But we will sort it by playing football, by sticking even more together, by fighting with all we have, by learning more than we have in each season we played before. That's the plan we have.
"We've learned a lot but in the moment a lot of people aren't happy for the results. I am responsible for that, 100 per cent. But we played still some pretty good stuff. You can forget that, we cannot because that is the start of changing things.
"Do you change a situation like we are in with bad football. I've never heard of that.
"We have to be dominant. Massive challenge, I'm ready, the boys are ready and we will give absolutely everything to sort it."
Klopp said: "Thanks for all the support. I'm very, very grateful for these kind of things. But they can really think about other things because no one has to worry about me."
Naby Keita will not be fit enough to face his former club despite returning to training this week. The midfielder has not featured since December's 7-0 win over Crystal Palace.