Van Dijk was formerly captain at Southampton prior to his £75m move, while Liverpool have lacked a regular skipper this season as a result of Jordan Henderson's injury problems - Philippe Coutinho, Emre Can and Simon Mignolet have all worn the armband.
Looking ahead, Thompson now sees Van Dijk in that role.
"Dare I say it, he looks a future captain material. The way he plays, with his presence, I'd like to think he could grow into that role in another 12 months," Thompson, a former Liverpool captain himself and three-time European Cup winner, told Sky Sports.
At 26 years of age and needing time to adapt to his new surroundings after a lengthy and unsettling transfer saga, Van Dijk is not the finished product. Thompson admits that, but has every confidence the Dutch international will blossom.
"He needs to be a better organiser, which will come with experience, because you need to have that ability to make sure everybody is organised, even when you're not having a good game yourself," the Soccer Saturday regular commented.
"I agree with the comment that we'll see the best of him next season when he's had a full pre-season after his injuries and his head not being in the right place for so long.
"I think he's absolutely fantastic when at his best, a Rolls Royce of a defender, reading the game very well and certainly a huge presence at 6ft4in with plenty of speed."