The Reds, along with Arsenal, were frontrunners in the race to sign Lemar with Manchester United and Barcelona watching from afar.
Monaco, however, were always clear that it would take a bid of around £100million to tempt them into selling.
The Liverpool Echo say that, in a bid to get around that, Klopp was willing to include Divock Origi.
But Monaco stayed true to their word and, consequently, Liverpool decided to abandon their pursuit, loaning Origi to Wolfsburg instead.
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Liverpool transfer news: Jurgen Klopp gave up on luring Thomas Lemar to Anfield
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Lemar also came close to joining Arsenal.
The 22-year-old was subject of a £90m bid after the Gunners were convinced Alexis Sanchez would join Manchester City.
However, Lemar ultimately chose to stay at Monaco for one more year.
And, having managed to find the net just three times, Liverpool now look set to look elsewhere.