Lallana was pictured in training at Melwood on Wednesday afternoon, having missed the previous four games with a thigh injury picked up on England duty in March.
This has led to speculation that the midfielder could return to Klopp’s squad against the Eagles.
“Unfortunately not too positive,” he told reporters of his squad’s health.
“Adam’s on a good way, yes, back on the pitch but with the coach. Nothing’s changed there.
“He should be involved next week in normal training. Wednesday, Thursday, something like this.
Work, sleep, repeat…. another good session completed. More still to do ? pic.twitter.com/jHiAdUTj7D
— Adam Lallana (@officialAL20) April 20, 2017
“In our situation we have to think about the player when he starts yesterday, that’s how it is. We will see.
“Danny Ings, after a long time in the gym, is running. He enjoys it a lot, but of course not even close to coming back.
“It was a very small ligament [problem for Klavan], really small but could not be enough time for Palace. It will be interesting.”
However, Klopp did confirm that Roberto Firmino “looks good,” after the Brazilian had battled with fatigue in recent weeks, joining the improving Lallana and Ings among the positives.