The Dutch international was heavily linked with a move to the Merseyside club in June, while their Premier League rivals Chelsea and Manchester City were also in the race for his signature. Klopp's side were believed to have won the race.
However, Liverpool were forced to end their interest in the defender and issued a public apology for "any misunderstanding" as there were suggestions the Saints were reportedly planning to report the Reds to the Premier League for alleged illegal approach.
Carragher remains hopeful Klopp's decision to not signing a central defender in the summer will convince him to go back in signing Van Dijk in January.
"We all know Van Dijk was the one [Klopp] wanted, so because he never got another centre-back, Klopp, now he's waiting until maybe January or the end of the season, I hope it's January," Carragher told the Liverpool Echo.
The former Borussia Dortmund manager has the services of Joel Matip, Dejan Lovren and Ragner Klavan as the recognised centre-backs in the squad. In addition to this, Joe Gomez can feature in the same position as well.
Carragher, who played in Liverpool's back four in the past, believes one of Matip and Lovren needs to be the third choice, while Klavan can be the fourth choice, if and only if, they can secure Van Dijk's services.
"It's a big concern (Matip and Lovren don't play enough), oh it's massive concern," the Anfield favourite explained.
"You always build any top team from your goalkeepers and your centre-backs. When me and Sami Hyypia played, we were fortunate as we never got injured and these two just get injured too much.
"Actually the stats aren't too bad when they play together, the record is pretty decent. Liverpool need quality. Basically one of those two needs to be third choice. You need a centre-back to come in and one of those to play with him, and (Ragnar) Klavan is your number four."
It is unlikely that Klopp will bring in a new defender to Liverpool in the mid-season transfer window. The German coach is of the belief that signing a defender will not be the solution to the problems at the Reds' back four.
The Mirror reports Van Dijk is valued at £70m ($93.5m) in January as he continues to attract interest from Liverpool and Pep Guardiola's Manchester City. Southampton manager Mauricio Pellegrino says he is unaware of the recent speculation linking the Dutchman to a switch to the Etihad.
"I don't think so, This morning I was with Les (Reed, Southampton vice-chairman) talking about that but nobody knows about that," Pellegrino explained, as relayed by Sky Sports.
"I know Virgil is an important player, there will be a lot of speculation around him like with other players because we have got a lot of important players in the squad.
"It is normal, I can't control this and I don't want to talk too much about that. Sorry, but tomorrow we have got an important game and the most important thing is these three points. I want him, there is a lot of speculation but I can't talk about that. I can't control information from outside."