Van Dijk swapped Southampton for Anfield at the start of 2018, and has quickly won a place in Liverpool hearts.
In just over 12 months the Netherlands international has helped whip the Reds defence into shape, turning a previously shaky unit into the Premier League's most formidable backline in 2018-19 which has conceded fewer than any other team so far.
That stability at the back has helped Liverpool build a four-point lead at the summit after 23 games; and Gillespie, who was a part of the last Reds team to win the English top flight in 1989-90, believes Van Dijk can make history - provided he turns his ability into titles.
“I’ve been asked a lot in recent weeks: Just where does Van Dijk stand, in relation to the likes of John Terry, Rio Ferdinand, Jaap Stam even, who are up there as the greatest centre-backs in Premier League history,” Gillespie explained to the Mirror.
“Honestly, ability-wise and technique-wise I would put him top. He’s got absolutely everything.
"But to become a living legend you have to be successful and win trophies and medals on regular basis... not just a one-off.
“Terry and Ferdinand, they had immense quality, but they also had longevity. Virgil, he’s probably ahead technically, but he needs to show it with trophies.”
The centre-back, 27, has made no secret of his admiration for Liverpool's rich history, revealing that he is in regular conversation with Reds great Kenny Dalglish.
And Gillespie believes that is a positive attitude which could help him go far at Anfield.
“I read those statements by Virgil van Dijk talking about former players like Kenny and Graeme Souness, Phil Neal, and it is clear he gets it," he added.
"They became legends because of their desire to win and what they achieved for this club, and I’m sure he feels the same.
“The current team has strong characters who can deliver trophies – and he’s definitely one of them.”