The Reds are yet to score against Sunday's visitors to Anfield in three attempts this season.
They were held to a goalless draw at St Mary's in the reverse fixture and lost both legs of the League Cup semi-final by a 1-0 margin.
Philippe Coutinho is still shaking off the dead leg he suffered in the 1-0 win at Watford, but will be in contention on Sunday.
“The injury is much better," he explained to Liverpool's official website.
"It was a hard tackle and what happened was a dead leg. It caused a lot of pain in the area I was hit, but the pain has reduced now.
“At the moment, I am feeling very well – the first couple of days were the most painful. Of course, when you have to come out of such an important game like this one, and when you leave that early, it frustrates you, but thanks to God it was nothing major – just a strong tackle."
On the 24-year-old's chances of making the encounter, Klopp added: “Phil yesterday was running for the first time – he looked a little bit like John Achterberg, but it’s good news! We cannot be 100 per cent sure because we always have to wait to see how he improves, but in this moment, it looks rather positive."
The Reds boss also offered updates on Sadio Mane and Danny Ings, who are both recovering from knee injuries.
“Mane is doing really well but there is no chance [for him to play]. He is around [Melwood] and everything has really went well so far.
“Ingsy is doing well, really well. He is enjoying training now for the first time with the ball a little bit and all that stuff. He is on a good way.”
Klopp stated "there is nothing new" to report on Jordan Henderson's foot issue. "That’s the thing with this [injury], you have to wait – especially he has to wait, which is quite difficult."
Neither team have any suspension issues to solve.
LIVERPOOL POTENTIAL STARTING LINE-UP
Coutinho may have to settle for a place on the bench given he has not yet been in full team training this week.
Adam Lallana, who replaced the Brazilian after sitting out five games due to a thigh problem, should be named in the starting XI.
If both can feature from the off, either Divock Origi or Lucas could give way.
It would be harsh on the latter, however, who has been brilliant in recent games.
SOUTHAMPTON TEAM NEWS
Southampton have altered their approach in fixtures against the Reds this season to be more obstructive than usual while looking to profit on the break - a template they are likely to use again at Anfield.
"We know Liverpool well now," Claude Puel said. "They are a great team, but we know we can play a fantastic game against them.
"It's an opportunity to show our level. We need to defend together and play our football when we have the ball."
Despite thrice failing to score against Southampton this season, Klopp pointed out that Liverpool did create multiple openings.
"I thought when we played them, we had our chances," he said. "It’s not like we had no chance against them or we struggled to make them.
"We have to defend with all we have, because that’s always the base for success, but when he create the chances, we have to use them."
There are no fresh injury worries for Southampton, who are still without Virgil van Dijk - a priority target for Liverpool this summer - through an ankle problem.
BEST OPTA MATCH FACTS
Southampton have won just one of their last five visits to Anfield in the Premier League (D1 L3), 1-0 in September 2013 with a goal from current Liverpool player Dejan Lovren.
Daniel Sturridge has played just 1316 minutes of Premier League football since Jurgen Klopp took over as Liverpool manager – just 22% of the possible minutes during this time and less than 17 other Liverpool players.
Divock Origi has appeared in 40 different competitive games for Liverpool so far in 2016-17; more than any other player in the squad. Over half of these (21) have come as a substitute, however.
TV COVERAGE & KICK-OFF TIME
Kick off is at 13:30 BST and the match will be televised live on Sky Sports in the UK.