Van Dijk was a notable absentee as Liverpool booked their place in the FA Cup fourth round with a 2-0 victory over Arsenal at the Emirates Stadium on Sunday.
It was confirmed before kick-off that the defender had picked up a virus and would have no involvement.
Liverpool have one more match to play - a Carabao Cup semi-final first leg clash against Fulham - before they go on their winter break and Klopp admits he doesn't know if he'll have his captain at his disposal.
But what he said afterwards will have Liverpool fans in stitches.
"Yes, I'm hopeful but I don't know..." Klopp said at a press conference. "It's difficult for Virg to look s*** but he did. So we sent him home again."
The Reds got the better of an Arsenal side that was wasteful in front of goal, with both Bukayo Saka and Kai Havertz missing good chances before a Jakub Kiwior own goal and Luis Diaz's stoppage-time strike sent the Gunners packing.
Things are looking very good for Liverpool, meanwhile, as they sit top of the Premier League and are still in all three cup competitions they've entered this season, although Klopp brushed aside any suggestion he might look at the result as a harbinger for the Premier League title race, or even a boost for the Reds' confidence in that chase.
He said: "No. I can't (read into it). It doesn't mean nobody can, but I can't. We don't have a problem with belief and confidence, we are not really bothered about the things you are talking about because we can read the calendar.
"We see it's January; it's cold outside, stuff like this, and you just have to stay warm and play as good as you can, and that's what we try."