The full Liverpool squad returned to Melwood from global duty to train late on Wednesday afternoon but it was without the German, who by then was already at hospital.
But the England global could make his return against the Saints.
"He was in a good way and then had a little setback".
In other injury news, Philippe Coutinho recovered well enough from an adductor problem to start for Brazil against England on Tuesday night.
Liverpool's first-choice right-back has been unavailable at all this campaign.
Southampton return to duty as they travel to face Liverpool on Saturday, in a game that could be very interesting.
Joe Gomez and Trent Alexander-Arnold have played in his absence this term.
Sadio Mane was forced out of Senegal's worldwide get-together early when the forward felt less than 100%.
Klopp's full squad returned to Liverpool's Melwood training ground from worldwide duty on Wednesday ahead of Saturday's home clash with Southampton but the players arrived with Klopp nowhere to be seen, as their coach had already been taken to hospital.
"So, in this moment all I can really say with certainty is that we will assess Sadio as soon as he lands back in the country and make decisions on what rehab and treatment gives him the best chance of being back fit and strong for the important fixtures we have coming".
Left-back Andy Robertson was back in training on Monday following Scotland worldwide duty.
Lallana took part in a behind-closed-doors friendly during the week, and if everything goes as planned, should be with the squad come Saturday.