The Portuguese attacker has been in the care of the medical team since sustaining a muscular injury in the Reds' 1-1 draw with Manchester City at the Etihad on November 25, coming off on the 54-minute mark against the champions.
Jota's affliction has seen the former Wolverhampton Wanderers man sit out Liverpool's last eight matches across all competitions, although he has been tipped to return before the New Year rolls around.
Prior to Liverpool's EFL Cup beating of West Ham United, assistant manager Pep Lijnders confirmed that Jota had restarted ball work at Kirkby, but the club were still adopting a cautious approach given their healthy selection dilemma in attack.
Jota was not given the green light to face Arsenal at the weekend, but speaking to the press ahead of Tuesday's trip to Burnley, Klopp did not explicitly state that the 27-year-old would not be available, in spite of his lack of team training.
"I don't want to rule out anybody now. We will see. He didn't train with the team yet and we have only one proper session, so it would be a surprise - but maybe I will be surprised and all fine," Klopp told reporters.
"I don't know. I saw him [on Friday]; he didn't train with the team but the session I saw was really intense, but I didn't ask yet who will be available. The medical department will tell me and then I will know more."
Jota's potential return at Turf Moor could be timely after Luis Diaz came off with a knee issue in Saturday's stalemate with Arsenal, although the Colombian is not thought to be nursing anything more severe than an impact injury and will likely be passed fit to make the squad.
However, Diaz was one of two Liverpool casualties in the 1-1 draw with the Gunners, as Kostas Tsimikas was sent tumbling into Klopp's path after losing a 50/50 to Bukayo Saka, and the Greece international will be out for the foreseeable future with a broken collarbone.
With Andrew Robertson also still recovering from his shoulder operation, Joe Gomez will most likely have to hold the fort at left-back single-handedly, unless academy product Luke Chambers is promoted to the Premier League ranks.
Klopp stressed the importance of wrapping Gomez in "cotton wool" while Liverpool have no senior left-backs available, adding: "Can Joey play all the games coming up now until - I don't know when - Robbo will be back? Kosti is now definitely out for a while, so I don't know, we have to see.
"I didn't have time now to think it through, but somebody will play the position I am pretty sure. It is for us obviously really tough, it's really tough. You have injuries sometimes and we accept all of them, but a broken collarbone is really bad because it just takes long.
"Same what Robbo had: a different injury but that takes long as well, how we all know. We all know we need luck in these moments to get through and now we will have Joey there and just have to make sure we put him in cotton wool and we have to make sure he is always ready for the games."
Ben Doak (knee), Joel Matip (ACL), Thiago Alcantara (hip), Stefan Bajcetic (calf) and Alexis Mac Allister (knee) are also in the Anfield infirmary, although the latter should also make his comeback within the next few weeks.
Liverpool's failure to beat Arsenal at the weekend meant that the Reds missed the chance to rise to the top of the Premier League table, but Klopp's troops will likely storm to the summit for at least two nights with victory over Burnley, as Aston Villa - who face Manchester United on Boxing Day - possess a far inferior goal difference.