The Uruguayan forward had to be replaced in the 59th minute of the game, where he had opened the scoring in the 10th minute.
Cody Gakpo added a second goal for the Reds seven minutes later, before the Magpies were reduced to 10 men after Nick Pope’s red ball for a handball outside the box.
But Nunez’s injury was a sour note for the Reds, who moved up to eighth with the victory making it back-to-back wins.
“The bad news is Darwin has something on his shoulder,” Klopp told reporters. “We will see.
“In that situation I dont know what he whistled there against him, to be honest.
“In the end, he had to go off with a shoulder issue. We need further assessment; I dont know in the moment.
“In the moment is it painful, but hopefully its just painful and not more.”
The injury comes ahead of Liverpool’s first leg of their Champions League last 16 clash with Real Madrid.
Klopp called the win “massive” for Liverpool’s season as they look to stay in contention for the top four.
Liverpool are now within six points of fourth-placed Newcastle with a game in hand.
“I don’t think we are in a position now where we can have a big mouth and say, ‘We are here and go again’,” Klopp said.
“It is obvious we are in a better place than we were a couple of weeks ago but there’s still space for improvement and we have to show consistency.
“So, next game Real Madrid, different competition, nothing to do with the game today. We have to recover and then go for that. After that it’s Crystal Palace. Palace is a tough place to go as well, so we have to keep going.
“When you are in the position we are in, there is no alternative to consistency to get out of it. And result-wise that was definitely good for consistency. And performance-wise, there was a lot of good stuff and areas we could improve.”