The Uruguayan played in the 5-2 defeat to Real Madrid after a pain-killing injection, but Klopp explained at Selhurst Park that he “couldn’t move the arm at all” the next day.
It was unclear how long Nunez would spend on the sidelines, and whether he would be fit to face Wolves at Anfield on Wednesday night.
But in his pre-match press conference, the manager confirmed that both Nunez and Ibrahima Konate were able to train “completely normal” on Monday.
“Now we have to see,” he told reporters.
“I think for Ibou he had no issues since he was injured, we just wanted to give him more training time before we include him again. That should now be fine.
“Darwin we have to see. It’s the shoulder.
“It looked absolutely OK yesterday, but I don’t know before [training on Tuesday]. The people will tell me how he reacted to yesterday.”
Klopp also gave a positive update on Diaz’s recovery from a long-term knee injury, though the “decisive information” is that he is yet to rejoin full training.
“Luis, I watched half an hour of the rehab session, out on the pitch individually, shooting, moving,” he explained.
“It looks all really good. But he was not in team training yet, so that’s probably the decisive information.”
“We didn’t have yet the full squad together on the training pitch, but last week, without Thiago and Lucho, there were 26 players out there on the pitch – not counting goalies,” he said.
“When they’re all in the squad is not too small. It’s not that we don’t have players.
“But we never had them together and most of the time we have six, seven, eight players out. Now we don’t have that amount and hopefully it stays like this.
“That would be massively helpful for this decisive part of the season.”