Uruguayan striker Nunez picked up a toe injury prior to the Reds' 3-0 win over Leicester City two Mondays ago, which forced him to sit out that success at the King Power.
Klopp subsequently revealed that Nunez had been nursing a "very painful" inflamed tendon, and he was nowhere to be seen in last weekend's 1-1 draw with Aston Villa either.
The 23-year-old was pictured back in training with the team earlier this week, though, and Klopp confirmed that he will be in with a chance of returning for the clash at St Mary's.
However, the Reds have also been handed two fresh defensive concerns, as Konate and Robertson will both need assessing due to illness and a groin twinge respectively.
"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," Klopp told reporters in his pre-game press conference. "Long-term [injured players] are still out, that's clear. The rest should be alright."
Liverpool's long-term injury list includes Thiago Alcantara (hip), Stefan Bajcetic (adductor) and Calvin Ramsay (knee), while Naby Keita has not seen the field since picking up a muscular injury in the latest international break.
Keita is one of four Liverpool players - along with Roberto Firmino, James Milner and Alex Oxlade-Chamberlain - confirmed to be leaving the club when their contracts expire this summer, and tributes were paid to the quartet in their recent stalemate with Aston Villa.
Firmino came off the bench to net a late equaliser in his farewell Anfield appearance last Saturday, which kept Liverpool's hopes of Champions League qualification alive before Manchester United met Chelsea in midweek.
A win for the Blues at Old Trafford would have taken the race for a top-four finish down to the final day, but Man United thumped their London counterparts 4-1 to ensure a top-four finish and consign Liverpool to Europa League football next season.
As a result, the Reds have nothing of note to play for on Sunday, and Klopp has hinted that he will make changes to his lineup, potentially offering some departing players their final starts in a Liverpool jersey.
"I cannot remember the last time we played on the last day and everything was sorted. Usually we play and everything is at stake," Klopp added.
"We want to play to win and that is the only reason we need. It makes sense to make changes. To get fresh legs in and play players that want to show something."
Following confirmation that Liverpool would be playing in the 2023-24 Europa League, Mohamed Salah claimed that his side had let their fans down in a strongly-worded social media post, and Klopp also responded to the Egyptian's message in Friday's press conference.