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The Reds won their second leg against Atalanta 1-0 in Italy but the damage was done in the first leg a week ago.
It means Klopp has just six matches left in charge, starting with this weekend's trip to Fulham, and his players must dust themselves down and keep their Premier League title hopes alive.
Liverpool go into this weekend in third, two points adrift of leaders Manchester City.
But Pep Guardiola's men are in FA Cup action, meaning Liverpool can go above them with victory at Craven Cottage.
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Sunday's encounter is the first of three away league games this week, with trips to Everton and West Ham to come.
It is said that the German side wants to keep him for at least another season, when they believe his valuation will be even higher.
Frankfurt currently sits sixth in the Bundesliga table and looks on course for a spot in next season's Europa Conference League.
Liverpool.com says: A central defender is expected to be high on Liverpool's wish list this summer, with Joël Matip likely to depart on a free transfer
It means any interested club may have to pay over the odds so the Reds may consider other options.
Mats Hummels throws shade at Premier League
It has been a disappointing week for English clubs after four exited European competition
Liverpool was joined by West Ham in being knocked out of the Europa League after losing to Bayer Leverkusen over two legs.
Manchester City and Arsenal had earlier been sent packing from the Champions League by Real Madrid and Bayern Munich, damaging hopes of a fifth qualification spot in the Premier League.
Real and Bayern will now face off in one semi-final, with PSG taking on Borussia Dortmund in the other
And Germany defender Mats Hummels has rubbed the struggles in the faces of the Premier League clubs.
The Bundesliga has often been referred to as a 'farmers' league' in the past, due to Bayern's dominance and Hummels posted on social media: "Good harvesting this week my fellow farmers,” alongside a handshake emoji and a German flag.
Liverpool.com says: English clubs have enjoyed great success in Europe in recent years but it has been a disappointing campaign this time around
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