The Netherlands international could be out for the rest of the season following the injury sustained during the Merseyside derby this month.
But his operation, which was carried out in London, has been described as a success, although Liverpool’s medical staff are not placing a timeframe on his recovery.
In statement the Premier League champions said: “Van Dijk will now immediately focus on the beginning of his rehabilitation with the support of the Reds’ medical department.”
Van Dijk suffered the injury in a challenge with Everton goalkeeper Jordan Pickford, who has been criticised for his part in the incident.
Liverpool No1 Alisson has added to that criticism, telling Sky Sports: “Every goalkeeper has a different way to play, a different way to move, but on that moment 100 percent it is not a necessary movement.
“But thinking what I have in my mind when I go for a situation, I go to finish the situation because if I go too soft I know the opponent can score.
“Everyone has their own way to go for the ball. I don’t agree with that way, everybody can see that but yeah it was just an unnecessary moment and movement.”
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