Thiago Alcantara missed Liverpool's 2-2 Premier League draw with Tottenham due to a suspected positive coronavirus test result, as Klopp was left without a raft of key players.
Fabinho, Curtis Jones and Virgil van Dijk were also sidelined after returning suspected positive results ahead of the midweek win over Newcastle United, and captain Jordan Henderson was another absentee due to illness at Tottenham, though he has tested negative for COVID-19.
Tyler Morton was handed a first Premier League start in midfield for the Spurs game, and he featured alongside James Milner and Naby Keita.
Liverpool's game was one of only four Premier League fixtures allowed to go ahead this weekend, with the other six matches postponed due to coronavirus cases among teams involved.
Indeed, Spurs were back in action for the first time since beating Norwich City on 5 December, after a COVID-19 outbreak in their camp.
Liverpool faces a Carabao Cup quarter-final against Leicester City on Thursday (AEDT), before Premier League clashes against Leeds United and the Foxes on 26 and 28 December respectively.
Klopp, though, wants the authorities to intervene and consider postponing – or even cancelling – some games.
"We play now on Wednesday, Sunday and Tuesday," he said. "That is impossible. We do not have the players.
"We could maybe have another case or two. We have to think about it, we cannot just push it all through.
"First of all, take away the second leg of the EFL Cup semi-final. Just play once; wherever you play it, I don't care.
"December 26 and 28 is really not possible. We might wake up tomorrow morning and might not be able to play anyway [due to more cases]. We would prefer to play, but we need some help. If we carry on, we cannot carry on as normal."
A meeting is due to take place on Tuesday (AEDT) involving Premier League clubs, as the situation intensifies and virus cases continue to emerge.
"The big thing we all have to talk about is: can we carry on? Do we carry on as usual? We need to have help from the schedule," Klopp said.
"If we don't play any more and have a break I am fine. But I know the problem. When do you want to play the games? It is not that easy to fit them in.
"We said before coronavirus that it was a bit busy, but people could not get enough football. Now it gives us a proper punch and tells you that you cannot carry on like this, and we have to consider what we do."