Salah underlined his importance to the Liverpool set up with a wonderful solo effort against champions Manchester City over the weekend.
The 29-year-old Egypt international’s current deal at Anfield runs out in the summer of 2023.
Although Liverpool have been able to agree new contracts with a host of their first team stars recently, reaching a consensus with Salah continues to drag on.
Salah has scored nine goals in nine games this term and Carragher has urged his former club to tie down a man he now considers a’ legend’.
"I don't think there's anyone playing better in the world or in Europe at the moment," Carragher said on Sky Sports. "His record at the start of this season has been absolutely outstanding.
"You can never really question him for what he's done at Liverpool - if at all - but right now is as probably as sharp and as good as I've ever seen him."
"His finish was world-class and he is world-class," added Carragher.
"I said this over the last couple of weeks - he is one of the greatest players to play for Liverpool. The club have lost Roger Hunt in the last few days, one of the greatest goalscorers the club has ever seen.
"Mo Salah is one of them, in terms of Ian Rush, Roger Hunt... Mo Salah is up there. He goes into a Liverpool greatest XI.
"You think of the greatest players Liverpool have had - he goes on that right side, there's absolutely no doubt about it.
"I know the situation Liverpool are in. They don't have the finances of Manchester City when you think of the wages they give out. But right now, Salah is playing as well as anyone in European football.
"Liverpool cannot afford to let his contract situation drag on, with the threat they could lose him in the next two years.
"This is a guy who really looks after himself, and you can see that when he takes his top off when he scores a goal.
"I'm not saying he's going to be like Cristiano Ronaldo when he's 35 or 36, still topping the goalscoring charts, but I think this lad's got a lot in him as he goes into his 30s.
"It's imperative that at his peak he's still scoring goals in the red shirt of Liverpool."
