The Colombia international has not seen the field for the Reds since sustaining the problem in a 3-2 Premier League defeat to Arsenal at the Emirates on October 9.
Diaz has missed Liverpool's last 10 competitive fixtures in all competitions, although he flew out to Dubai with the rest of the squad for a winter training camp.
Liverpool initially received a significant boost when Diaz returned to training with the team in the Middle East, but he soon suffered a setback which scuppered his recovery.
Diaz was not able to take part in Sunday's 3-1 Dubai Super Cup defeat to Lyon, and there are fears that he could be sidelined for up to three months should he require surgery.
Speaking to LFCTV after his side's loss to Lyon, Klopp offered a pessimistic update on Diaz's condition, although he did not confirm whether the 25-year-old would need to go under the knife.
"He is not in the best possible place [mentally], that is clear. It was a big disappointment for all of us and him as well. It was a nothing situation in training, honestly," Klopp said.
"And he felt something, he didn't feel it a lot the next day but we wanted to be really cautious. So we said: 'come on, let's have a proper look' and then the news came which was a proper smash in the face, but that's it now."
Diaz hit the ground running at Liverpool after making the £37m switch from Porto in January, and he already has 10 goals and eight assists under his belt from 38 games for the Merseyside giants.
The South American scored four goals and set up three more in 12 outings this term before sustaining his injury, and he is joined by Arthur and Diogo Jota in the Anfield treatment room.
On a brighter note, Naby Keita was able to return for Sunday's defeat to Lyon, where Liverpool took the lead inside 40 seconds through Fabio Carvalho before Les Gones turned the game on its head.
Two goals from Alexandre Lacazette either side of a Bradley Barcola effort saw Lyon march to a comfortable win, and Laurent Blanc's side also claimed the additional point in the penalty shootout after Calvin Ramsay's spot kick was saved by Anthony Lopes.
Klopp introduced a wealth of academy graduates in the second half and affirmed that there were some positives that he could take away from the defeat, adding: "The plan is we want to chase it and really be ready immediately, that's why we have to use this time now again to train and give the boys proper intensity in training, which again is why I am really happy with big parts of the first half, knowing that it was in the right direction.
"That was not always the case with the summer games. I saw a lot of things tonight that I wanted to see and I am positive but of course we play some tough, tough teams to come.
"The start of the game was really good from us, I liked it a lot. Some parts in the first half, I saw what we have been working on in the training ground so far.
"We were really aggressive, well-connected, could have scored another goal but we conceded with a misunderstanding between Robbo [Andy Robertson] and Caoimhin Kelleher.
"Every ball we lost was a big chance on the counter and that, of course, is not how it should look. But that's a learning curve and we will use this to learn from."
Liverpool have until Friday to recover before they take on AC Milan, who first pit their wits against Arsenal in Tuesday's Dubai Super Cup contest.