As things stand, RB Leipzig star Keita will join Liverpool on July 1 after a £58million fee was agreed last summer.
After losing Philippe Coutinho to Barcelona last week, Klopp has made it a priority to bring in a replacement.
As far as Keita is concerned, the German tactician wants to get him in this month so as to aid the Reds with their push for a top-four spot in the second half of the campaign.
And according to the Daily Mail, Liverpool’s sporting director Michael Edwards will hold talks with his opposite number Ralf Rangnick to speed up Keita’s arrival.
Reports emerged yesterday suggesting the Reds had been quoted a premium fee of £13million which has to be paid up front now if Keita were to leave this month.
But the Mail claim the Merseyside outfit are yet to be informed of the price.
RB Leipzig boss Ralph Hasenhuttl has suggested Liverpool have no chance of getting Keita this month.
He said: “The current situation is that nothing has changed.
“We do not have a reason to sell Keita any time before the end of the season. That’s it.
“He will play at the weekend against Schalke and hopefully he does well. That’s the only thing I really care about.
“I think that Naby wants to finish the season in a good way in Leipzig. He wants to leave a good impression.