The defender - who will wear the vacant No.4 shirt - will become the Reds' most expensive ever signing and joins when the January transfer window opens next Monday (January 1).
The Saints have finally agreed to sell the want-away 26-year-old after a poisonous summer saga saw the Dutchman denied his desired move to Merseyside despite submitting a transfer request.
The agreed fee sees Van Dijk become the world's most expensive defender, surpassing the previous record £54 million fee - if all add-on clauses are met - that Manchester City paid to sign Kyle Walker from Tottenham Hotspur last summer.
The two Premier League clubs have been in negotiations this week and Liverpool have acted quickly before the opening of the January transfer window following reports City had made Van Dijk their top priority, while Chelsea were also said to have shown interest.
It has been suggested that Van Dijk has shunned interest from City in order to complete a long-awaited and coveted move to Anfield, where he will sign a deal until June 2023, reports say.
Jürgen Klopp has finally landed the man he made his only central defensive target in the summer despite Liverpool's obvious need for strengthening at the back.
But his stubbornness and patience has paid off with Liverpool finally acquiring the German's first-choice option, reports suggesting ex-Celtic defender Van Dijk will join on a deal worth around £180,000-a-week.
The German's latest signing took to social media to declare himself "delighted and honoured" to join "one of the biggest clubs in world football."
He added: "I can't wait to pull on the famous red shirt for the first time in front of the Kop and will give everything I have to try and help this great club achieve something special in the years to come."
Van Dijk will not be able to register as a player until January 2 meaning he will miss the trip to Burnley on Monday night, their first game of the new year.
Van Dijk to Liverpool a done deal at long last
The move brings to an end an exhausting and complicated tapping-up saga which rumbled on for months and, much to the joy of neutrals as well as fans of both clubs, ends the daily rumours linking Van Dijk with a move.
Van Dijk made it clear of his intentions to join Liverpool in the summer. While no official bid was submitted after Southampton's rage at leaked reports suggesting a deal was all but done, the Merseyside club had made clear their willingness to spend £60 million-plus to sign the player.
Southampton reported an illegal approach with Van Dijk having spoken to Liverpool before a bid had been accepted by his club, forcing an embarrassing public apology in which Liverpool insisted they had "ended any interest in the player."
Van Dijk handed in a transfer request and refused to play at the start of the season as he looked to force Southampton's hand, but remained on the South Coast with new boss Mauricio Pellegrino tasked with the job of reintegrating him into his first-team squad.
Van Dijk made his first appearance of the season in late September and has since featured in 12 Premier League games, but was left out of the squad that travelled to Tottenham Hotspur for a 5-2 defeat on Tuesday having not played against Chelsea or Huddersfield Town.
But Southampton's dire form is believed to have influenced boss Pellegrino's decision to allow Van Dijk's sale, with the Spaniard keen to try and improve his struggling side with a raft of mid-season signings.
The funds from the Netherlands international's departure will undoubtedly aid the former Liverpool defender's hopes of strengthening his options as he looks to lift them well clear of relegation - only two points separating them from the bottom three.
It also sizeably improves Liverpool's choice at the back, with Klopp having had to start the inconsistent duo of Dejan Lovren and Ragnar Klavan together regularly due to Joël Matip's injury tendencies.
Klopp is reportedly wholeheartedly convinced that Van Dijk - the joint-second most expensive transfer in the history of the Premier League behind only £90 million midfielder Paul Pogba - is the man to shore up a Liverpool back-line that has leaked 23 goals in 20 top-flight games - seven more than any other side currently in the top four.
Not just that but Van Dijk's arrival arguably represents another big statement for the club after sealing a £48 million deal to bring in Naby Keïta from RB Leipzig next summer back in August having also held off Barcelona's attempts to prise away playmaker Philippe Coutinho several months ago.
As of yet there have been no suggestions that Klopp will look to recruit more new additions with Liverpool's Champions League involvement - which Van Dijk will be eligible to play in - resuming in February as well as their continued top-four battle.