The Egypt international put the ball firmly in the club's court after scoring and setting up two goals in Liverpool's 3-0 victory at Manchester United on Sunday.
"Nobody in the club talked to me yet about contracts so I was just like, 'OK, I play my last season and see at the end of the season'," he told Sky Sports.
Salah is 32 but Carragher does not believe that will be a barrier to the forward and therefore should not be for Liverpool.
"I'm not sure it will be his last year. I think Mo Salah is a little bit like (Cristiano) Ronaldo," he told The Gary Neville Podcast.
"Most footballers think 35 is the time to finish. I think Mo Salah will be looking at his late 30s and winding down then in his head.
"He will be looking at breaking every record he possibly can, either in the Premier League or Liverpool.
"I think he is aware of his status and I find it very difficult to see Mo Salah in the Saudi league, for instance, next season. He is playing that well at one of the biggest clubs in the world."
In July 2022 Salah signed a new three-year deal, reportedly making him the highest-paid player in the club's history on around £350,000 a week.
Liverpool reportedly rejected a £150 million offer from Saudi's Al-Ittihad for Salah last year.
The Egyptian has scored 214 goals for Liverpool since joining in 2017, putting him fifth on the club's all-time scoring charts.
New Liverpool boss Arne Slot refused to be drawn on the issue of Salah's future after the win against United.
Asked about the player's comments, Slot said: "It's a lot of 'if'. At this moment he is one of ours and I am really happy with him being one of ours and he played really well.
"I don't talk about contracts from players but I can talk for hours about how Mo played today."