The Netherlands international has been mooted as a top target for the Reds before and during the summer transfer window.
Liverpool were keen at one stage, with the club forced to apologise to the Saints after being accused of making an illegal approach.
They stated at that point that any interest had been shelved, and Klopp insists that remains the case despite admitting to still being in the market for additions ahead of the August 31 deadline.
He told reporters when quizzed on the Van Dijk rumours: “We have four centre-backs. I don’t think we need more.
“Just at the moment I’m fine. The transfer market is open until the 31st of August.
“We cannot force things and we would never get a centre-back just because someone is available.
“Look out there and tell me five that would make us stronger. Then you win a prize! It’s difficult.
“I’m happy. You can imagine we still work on different things and we will see if it works.
“I am here to work with this squad and not just always thinking, ‘But if we don’t get this or this’.”
Liverpool already have Dejan Lovren, Joel Matip, Ragnar Klavan and Mamadou Sakho on their books.
They also have the versatile Joe Gomez, who is capable of operating at full-back or in the centre of a defensive unit.
France international Sakho is expected to be moved on, with could alter Klopp’s stance, but for now Van Dijk is generating stronger links to reigning Premier League champions Chelsea than title hopefuls at Anfield.