Klopp's side rejected a £150m bid for the Egypt international from Saudi Pro League club Al-Ittihad last week, with the Reds insisting the 31-year-old is not for sale.
With the transfer window in Saudi Arabia not closing until Thursday, Liverpool are bracing themselves for more bids in the coming days.
But after watching him score one goal and play a part in the others as the Reds beat Aston Villa 3-0 in the Premier League on Sunday, Klopp insists Salah is focused only on doing his best for the club.
"I never had any doubt about his commitment to this club," the German told Sky Sports.
"You can't imagine how much fuss the world has made but how calm we are with it. He is our player and wants to play here."
That belief in Salah's desire to remain at Liverpool appears to be felt in the dressing room.
Summer signing Dominik Szoboszlai, who scored his first goal for the club after just three minutes against Villa, said: "You know it is football, everyone is talking. We are really happy that he has stayed.
"We are of course speaking between each other but he wants to stay, he wants to be here and be with us.
"We are really happy - we need people in the team like him."
It is understandable Szoboszlai and his team-mates would be keen to emphasise Salah's importance to the team.
With his goal against Villa, he has has now scored or assisted in 10 consecutive appearances in the Premier League - the longest run since the frontman himself went on a 15-match streak between August-December 2021.
Salah, who has two goals in four games so far this season, has not scored fewer than 19 in a single season since moving to Anfield in 2017. He also has 188 goals for the club, putting him fifth in their all-time list.
While the Reds are being offered a significant sum of money for a 31-year-old, that is a lot of goals from one individual to try to replace.
"He has unbelievable numbers but it's not a surprise that he has the numbers," added Klopp.
"He didn't look for a second like he thought about anything else other than about Aston Villa, being involved in all the goals."
There was certainly no question about Salah's commitment with his performance against Villa.
But every player has their price and, should Al-Ittihad returned with improved bids in the coming days, it could prove to be a real test of Liverpool's resolve.