The Reds have tracked the German international for some months with Klopp a big fan of his compatriot, who has 31 goals across all competitions this season.
But Chelsea are said to have now reached an agreement with the 24-year-old to bring him to Stamford Bridge this summer, according to the Daily Telegraph.
He will sign a long-term contract worth around £200,000 a week and an agreement between Chelsea and Leipzig is expected imminently.
Liverpool refused to meet Werner’s release clause, which stands at around £53million, while their Premier League rivals do not share the same concerns.
Chelsea have only signed two player in the past 12 months due to a FIFA transfer embargo, bringing in Mateo Kovacic from Real Madrid on a permanent basis for a fee in the region of £40m and signing Hakim Ziyech from Ajax for an initial £33.3m.
But after failing to sign a striker to offer back-up for top-scorer Tammy Abraham in the January window, it appears Frank Lampard is closing in on the acquisition of Werner as he further bolsters his attack.
The Leipzig frontman has 25 goals and seven assists in just 29 Bundesliga matches this term, including four goals in as many games since the German league restarted amid the coronavirus pandemic.
He would represent a marquee signing for Chelsea, particularly at this time, when their spot in next season’s Champions League is far from guaranteed and given the rival interest of Liverpool and Manchester United.
With the Premier League set to restart on June 17, Chelsea occupy fourth spot but Manchester United – in fine form before the enforced break – are just three points behind.
Wolves and Sheffield United are five points adrift of Chelsea, the latter with a game in hand, while Tottenham cannot be discounted due to Harry Kane’s return, despite a seven-point gap.
While Werner has publicly hinted at his desire to move to Merseyside on several occasions, it appears Liverpool’s dithering over a deal has put him off.
His current release clause expires on June 15, with it dropping to just £36m in 12 months’ time.
The English champions elect had hoped that letting his clause to expire would allow them to bid below that fee later this summer, given the current climate of the transfer market after the coronavirus crisis, which means such big fee will be a rarity in a huge contrast to recent windows.
It remains to be seen whether Liverpool will make a last-ditch attempt to beat Chelsea to the deal, but the Telegraph add that insiders at the London club are confident of wrapping up his arrival.
What Werner’s move means for Abraham, the 22-year-old who currently leads Lampard’s line, remains to be seen though the German can play off the flanks.
Abraham has only played as a centre forward in west London and has 15 goals in his first season as a regular in the senior squad, including 13 in 25 league matches.
But given Lampard’s ambitions of restoring Chelsea as a major challenger in domestic and European competitions, the addition of Werner – added to that of Ziyech – certainly boosts their chances of doing so, while also ensuring Liverpool do not strengthen further.
Speaking about Werner’s future earlier this week, Leipzig boss Julian Nagelsmann admitted the player will decide where he plays next season.
He said: “There is nothing we can do about it. He knows what he has in Leipzig and also what he has with me.
“I show him that — he gets many chances to develop himself. I don’t know if it will help him that much if I tell him every day.
“A player has to feel that he can get better. If he doesn’t feel that, it wouldn’t do any good if I kept on telling him to stay here.
“In the end, it is his life, his career, he can decide that. I can’t think of anything new to say to Timo.”