Luis Diaz could make himself available for selection this weekend after a positive update regarding his father's kidnapping in Colombia.
Both of the Liverpool winger's parents were captured last weekend at a petrol station in the town of Barrancas before his mother was rescued by police, with his father Luis Manuel Diaz yet to return home.
Colombian guerillas the National Liberation Army (ELN) have now been identified as behind the kidnapping and vowed to release him "as soon as possible".
Jurgen Klopp said on Friday that the door is open for Diaz, 26, to play for Liverpool at Luton this weekend if he is ready to do so.
"If he feels right, he will be will us," said Klopp.
"You can see he didn't sleep a lot.
"The news from Colombia gives us a little bit of hope. I cannot say what we will do. It's up to him, if he makes himself available or not."
Authorities have been scouring forest areas near the border with Venezuela in the search for Luis Manuel Diaz and offered a £40,000 reward for information.
Diaz has not played for Liverpool since the incident and a shirt bearing his name was held up by teammates following the first goal of the win over Nottingham Forest last Sunday.
"We cannot force the Luis situation," Klopp added. "It is not in our hands. We're all waiting together for good news."
It had been claimed by ELN representative Juan Carlos Cuellar on Thursday night that Diaz's father will be released "as soon as possible", at a community meeting seen by Reuters.
The Colombian government's peace delegation had earlier identified the ELN as the perpetrators and stated: "We demand that the ELN releases immediately Mr Luis Manuel Diaz and we say as of now that they are entirely responsible to secure his life and integrity."
