Egyptian star Salah's current contract is set to expire at the end of the season, and the world-renowned attacker recently stated that a new deal is "far away" from being finalised.
Klopp, who was at the helm at Anfield when Salah signed for the club from Roma in 2017, has since moved on to pastures new in a unique role as Red Bull's global head of soccer, but still regularly fields questions on his time with the Reds.
When asked about the situation during his first official press conference in the job, Klopp told reporters: "I hope he stays - he's the biggest striker Liverpool had in modern times.
"Obviously, they had other good strikers, really good strikers. But fantastic player, fantastic human being, outstanding athlete, the best ambassador your country (Egypt) could have.
"He's fantastic in all departments, so I hope he will stay at Liverpool."
Salah made an immediate impact upon returning to England, having endured a fruitless stint under Jose Mourinho at Chelsea a few years prior. He scored an impressive 44 goals across all competitions in his debut season.
The pair went on to enjoy significant success together, winning numerous trophies, including the Premier League, Champions League, FA Cup, Club World Cup and League Cup.
No player throughout Klopp's entire managerial career has scored more goals (211) or provided more assists (88) than Salah.
This season, the world-class forward has excelled once again under new boss Arne Slot, with his 18 goals and 13 assists in 19 Premier League appearances propelling Liverpool to a seemingly comfortable lead at the top of the table. He is also the current top goalscorer.
Speaking earlier in January, Salah told Sky Sports that he "really wanted to win the Premier League" and suggested it was his last year with the club.