They added that the other members of the Egyptian national team had tested negative.
Salah had also attended a ceremony organised by the Egyptian Football Association (EFA) in Cairo to celebrate his achievement of being the first Egyptian to win the Premier League and the Champions League.
The result of the 28-year-old winger's first test was positive, but the EFA made a decision to conduct a second test to confirm the infection, according to the statement.
He was wearing a mask that was pulled from the bottom of his nose and around his chin in several photos taken at the ceremony, which was also attended by his global colleague and Aston Villa star Mahmoud Trezeguet.
Salah underwent a routine test ahead of the Egyptian team's first-leg clash with Togo on Saturday in the 2021 African Nations Championship qualifiers.
Medical swabs on the national team "showed that our global player, Mohamed Salah, Liverpool star, was infected with the coronavirus", it said in a tweet, adding he was "not suffering from any symptoms".
But the loss of Salah, who is joint leading scorer in the league this season with eight goals, eats further into Liverpool's squad depth that is being tested to the limit.
"Salah is keeping a high spirit and is not showing any symptoms of the virus".
"A special room has been prepared for isolation and another room for medical follow-up, while the team doctor, Dr. Muhammad Abu Al-Ela, communicates with Liverpool doctor Jim Moxon".
If he is sidelined, it would be a blow for Premier League champions Liverpool, who were deprived of Senegal global forward Sadio Mane earlier in the season when he too tested positive for coronavirus, although he has since returned to the team.
It's yet another blow to the Premier League champions who saw Joe Gomez ruled out for the majority of the season after suffering a serious knee injury in training with England this week, the centre-back joining a growing list of absentees that also includes Virgil van Dijk, Trent Alexander-Arnold, Fabinho and Alex Oxlade-Chamberlain.
Gomez is set for a lengthier spell on the sidelines.