The defender was left out of the 2-2 draw at home to Newcastle on Sunday (AEST) after it emerged UEFA were investigating a possible anti-doping violation.
PA Sport understands the France international failed a drugs test - believed to be associated with a so-called 'fat-burning' supplement - after Liverpool's UEFA Europa League second-leg tie at Manchester United last month.
Sakho has not been suspended either by the club or UEFA, but it is understood that after discussions between the club's owners Fenway Sports Group, Klopp and the player himself it was decided the centre-back should not play in the immediate future.
The defender is likely to request his B sample be tested, with the deadline for that request set for Tuesday.
"I'm really limited what I can say," said Klopp.
"There's is an official statement and there's nothing else to say.
"It's not the most easy situation that's for sure, but maybe we can say more when we know more maybe next week."
The club have been pro-active in their response to the alleged doping violation, consulting fully with UEFA and the Football Association, and, while Sakho is technically still available to play, it was felt the best course of action was to withdraw the player from selection and it is understood Sakho himself was fully behind the decision.
There will be no disciplinary action for the club to face - that can only kick in if two players in same season are found to have had doping violations - and there is absolutely no risk to their continued participation in the Europa League.
In a statement UEFA said: "UEFA would like to confirm the information communicated by Liverpool FC regarding an adverse finding in a doping test of their player Mamadou Sakho conducted at the UEFA Europa League match between Manchester United FC and Liverpool FC on March 17 2016.
"The player and the club have received all the pertinent information and have until Tuesday to request the analysis of the B sample as well as to provide explanations for the presence of a prohibited substance in the player's body.
"There are no disciplinary proceedings opened at this stage.''
Michele Verroken, director of Sporting Integrity and formerly in charge of anti-doping in the UK, advises athletes against using any substances described as 'fat-burners'.
"What is causing that fat to burn is that these supplements contain a form of stimulant,'' she said.
"They are not regulated products. It's just too big a risk. I warn against any weight-loss products. It's probably going to be a prohibitive supplement.
"It's really not worth it. It's a huge risk."
Verroken said that, if an athlete could show that they did not intend to improve performance, they could get a lesser sanction.
"But these are strict liability offences and have to go through the due process,'' she added.
Sakho's fellow central defender Kolo Toure was suspended for six months after failing a drugs test while at former club Manchester City in February 2011 having tested positive for endroflumethiazide, a weight-loss drug contained within "water tablets" recommended to him by his wife.