The Dutch defender joined the Anfield club from Southampton for a world-record fee of £75million.
The Daily Mail says as he was putting the finishing touches to his move, United made an approach, only to discover they were too late.
United did actually throw their hat into the ring and made an enquiry for Van Dijk.
They wanted to know the terms of his deal last December but the question was asked on the day Van Dijk was undergoing his medical at Anfield. By then it was far too late.