The 54-year-old German goes into a period of self-isolation and will miss the trip to Chelsea on Sunday.
Klopp is the fourth Premier League manager in recent times to test positive for the virus after Steven Gerrard, Patrick Vieira and Mikel Arteta.
The club said in a statement: "Jurgen Klopp will miss Liverpool's Premier League meeting with Chelsea on Sunday after returning a suspected positive Covid-19 test result.
"The Reds manager, who reported mild symptoms ahead of the fixture, is now isolating.
"Assistant manager Pepijn Lijnders will therefore lead the team for the 4.30pm GMT kick-off at Stamford Bridge.
"Testing of the entire first-team set-up has revealed no further positive cases within the playing squad in addition to the three confirmed by Klopp on Friday.
"Three backroom staff members have returned suspected positive results, however."
Liverpool will look to get back to winning ways at Stamford Bridge after losing their last Premier League game at Leicester City.
The Reds are now 12 points behind leaders Manchester City, who beat Arsenal 2-1 on Saturday, but have two games in hand on the champions.
Chelsea and Liverpool drew 1-1 at Anfield in the first half of the campaign.
