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Now, he could be on the move as a replacement for another of the manager's former lynchpins at Borussia Dortmund.
For its part, Liverpool is not thought to be keen on losing Van Dijk, with incoming manager Arne Slot making the defender's future his 'top priority'.
But the defender only has a year left on his contract, and according to the Mirror, Borussia Dortmund is leading the interest for his services.
With Mats Hummels out of contract even sooner, and set to depart this summer as things stand, Dortmund is keen to 'cover its bases' with Van Dijk.
The Dutchman is eager to carry on at the highest level, and a move to Klopp's old club would likely be one of the more appealing transfer options from this perspective.
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Hummels was a stalwart under Klopp at Borussia Dortmund.
Given the contract situation and the 32-year-old's known disappointment at Klopp's departure, clubs are taking a look.
However, Slot wants to keep Van Dijk for at least the year that is left on his contract.
While Slot no doubt wants to work with Van Dijk, who remains one of the best in the world, the decision over whether to negotiate a renewal probably lies mostly elsewhere.
Incoming sporting director Richard Hughes will also have a say, and it remains to be seen what kind of stance he takes.
But if no new contract is forthcoming for Van Dijk, it wouldn't be a huge shock to see him consider his options, with a transfer giving him time to get his feet under the table at a new club while still somewhere close to his prime.
Obviously, Liverpool fans will echo Slot in hoping Van Dijk stays put for at least another year, and hopefully longer.