Austin played with Van Dijk at Southampton, at a time when several European and Premier League giants were keen to secure the services of the Netherlands international.
And Austin, who is currently on-loan at Queens Park Rangers, told talkSPORT that United snubbed his former teammate, who has gone on to be one of the finest centre-backs the division has seen.
The Red Devils were reported to be tracking the Dutchman in the summer of 2017, as Jose Mourinho looked to bolster their backline but eventually decided on Benfica defender Lindelof for around £40million.
"I remember at the back end of his Southampton career, we were sitting on the bench together at Chelsea away. I said to him 'come on Virgil what's going on, surely Manchester United must have been in for you?', Austin said on talkSPORT Drive.
"They're a massive club in England. He said 'do you know what Chaz, in the summer it was between me and Lindelof, and they went for Lindelof'.
"Nobody knows that. He told me on the bench and I couldn't believe it. I was gobsmacked."
The former Southampton striker also says Van Dijk's transfer didn't go down well with then manager Mauricio Pellegrino, and he did not get the best send-off by the club.
"The manager was Pellegrino at the time and in the summer he was looking to go," he added.
"I remember at the time he got everyone on the pitch, including Virgil, and just came out and said 'look lads Virgil wants to go, the club aren't allowing him to go, he said he's not going to play so he's not going to train with us.'
"In front of everybody, including Virgil. He was gobsmacked, there was no heads up from the manager, nothing."
Van Dijk of course ended up at Anfield in January 2018 for a then world-record £75million, and has become one of the Reds' most important players and one of the world's best centre-backs.
Since his arrival on Merseyside, the Netherlands captain has won various honours including the Champions League and Premier League.