The Dutchman became the world's most expensive defender last month after sealing a much anticipated £75m switch to Anfield from Southampton.
Thompson, who captained Liverpool to European Cup success in 1981, believes Van Dijk has all the attributes to follow in his footsteps.
"Dare I say it, he looks a future captain material," he told Sky Sports.
"The way he plays, with his presence, I'd like to think he could grow into that role in another 12 months.
"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."