Demirbay was on the losing side as his Hoffenheim side were beaten 6-3 on aggregate by Liverpool in their Champions League play-off.
At the end of Wednesday second leg clash, Demirbay was spotted in an embrace with Klopp on the pitch.
Liverpool have had a difficult transfer window with moves for top targets Naby Keita and Virgil van Dijk failing to materialise.
And with less than a week until the end of the transfer window, Klopp faces a race against time to bolster his squad.
But according to German newspaper Bild, the Reds boss is keeping tabs on Demirbay and the player has been made aware that he is a Liverpool type player.
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Klopp knows all about Demirbay after the pair were together at Borussia Dortmund.
And ahead of the play off tie, Klopp praised the 24-year-old saying: “I like Kerem.
“He is a great player with a good development.”
The report says Demirbay is unlikely to join Liverpool this summer but there is an exit clause in his contract.