The Reds confirmed on Monday (AEDT) the Netherlands international had damaged ligaments in his knee and is likely to miss most of the rest of the campaign.
Van Dijk was subjected to a wild challenge by goalkeeper Jordan Pickford in Saturday's 2-2 Merseyside derby draw with Everton and limped off after 11 minutes.
The 29-year-old has since vowed to come back "better, fitter and stronger" after surgery but, while confirming he will be out for a "pretty long time", Liverpool boss Klopp would not be drawn on an exact date for his return to action.
"He will now be out for a while, that's clear," he told Liverpool's official website. "We don't want to set any timeframe, to be honest, because each player is different and these possible timeframes are always for the x, y, z player. Virgil is Virgil and we don't want to set any timeframe.
"But he will be out for a pretty long time and that's what the situation is. I expected it pretty much after the game, especially when I saw the challenge back. I saw it the first time only from the bench, and when I saw it back I was pretty clear that he will be out for a long time.
"So, we feel now in this moment extremely, extremely [sad] for him because most of us were in a similar situation and we know that these situations are absolutely rubbish.
"You don't need it but you have to get over it and Virgil will get over it, 100 per cent, and today is already the first day, if you want, of the recovery.
"We already go in the right direction. So, see what you get and then make the best of it and that's what we do now."
Klopp says he understands what Van Dijk is going through after suffering a similar injury during his own playing career and insists the club will do everything it can to support him.
"It's always the same; players love to be on their own to deal with it to begin with," he added. "I have been in the business long enough to know that. I was injured myself; I had exactly the same injury and you're not pretty chatty.
"We are in contact, of course. We are there for him. He knows that and we will wait for him like a good wife is waiting when the husband is in jail and will do all we can to make it as easy as possible for him."