Van Dijk, 29, will be forced to miss a large portion of the 2020-21 campaign after being caught on the receiving end of a nasty challenge from Jordan Pickford in the 2-2 draw with Everton on Saturday.
"A few things are clear," Klopp told the club's official website. "So he now will be out for a while, that's clear. We don't want to set any timeframe, to be honest, because each player is different and each individual is different and these possible timeframes are always for the x, y, z player [but] 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. That's what we know since… I expected it pretty much on Saturday after the game immediately, 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 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. 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 go already in the right direction and so that's OK and the rest is football and life, like always a challenge, so see what you get and then make the best of it and that's what we do now."
Klopp said that Van Dijk has preferred to spend time alone to come to terms with the nature of his injury, but stressed that both he and the club will be there 'like a good wife' to support the Dutchman.
"It all will start but we are in contact, of course, we are in contact. 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, meanwhile, all what we can do to make it as easy for him as possible."