The Uruguay forward has missed the Reds’ last two matches due to a toe injury but could be in contention to feature at St Mary’s Stadium on Sunday afternoon.
Speaking on Friday morning, Jürgen Klopp also reported news on Ibrahima Konate and Andy Robertson.
The manager told the media: “He [Nunez] trained yesterday, fully. Ibou was ill yesterday. Robbo felt something in his groin yesterday, he had a scan, that was fine but we anyway have to see how we can deal with that.
“Long-term [injured players] are still out, that’s clear. The rest should be alright.”