The Reds have been heavily linked with a move to sign the Netherlands international in the summer transfer window but a deal has failed to materialise so far.
Indeed, Liverpool were forced to issue a statement apologising to Southampton about their pursuit of Van Dijk and officially ended their interest in the former Celtic man.
However, the Merseyside outfit have still been linked with a summer deal to sign the 26-year-old centre-half ahead of the start of the 2017-18 Premier League season.
Former Liverpool captain Souness is desperate to see his former club complete a deal for the Southampton centre-half to bolster Klopp’s defence ahead of the new campaign.
“A new centre-half, that would have been the main one for me,” Souness told talkSPORT.
“If I was manager of Liverpool at this time I would have pushed the boat out and done whatever you had to do to get the boy Van Dijk.
“It looks like Southampton are digging their heels in, but Southampton do sell players, they have done consistently over the last few years.
“I’m an enormous fan of Van Dijk, I would love for him to go to Liverpool.
“When a player makes things look easy, that tells you they’re really good at what they do.
“Whoever gets him would improve them greatly, and it’ll be a shame if it’s not Liverpool.
“But, it may still happen, we don’t know.”
Liverpool finished in fourth place in the Premier League table to secure a return to the Champions League in their first full season under German coach Klopp.
The Reds have signed AS Roma winger Mohamed Salah and England Under-20 international Domimic Solanke in the summer transfer window so far.
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