The 33-year-old Egyptian forward, who signed a two-year contract extension in April, has been way below his record-breaking form from last season's campaign and has gone seven matches without a non-penalty goal.
Salah squandered a golden opportunity to score in Liverpool's 2-1 defeat by Manchester United at Anfield on Sunday and Carragher said he should not be one of the first names on the team sheet like captain Virgil van Dijk.
"Liverpool have got two away games - in the Champions League at Frankfurt and then they go to Brentford. I don't think Salah should start both of those games," Carragher told Sky Sports.
"He should always start at Anfield because Liverpool will be on top, around the edge of the box and more often than not he scores in those situations today.
"But I do think in the away games and helping your full-back, I don't think Salah should be starting every game right now, certainly away from home, with the form he's in."
Liverpool came into Sunday's game on the back of league defeats by Crystal Palace and Chelsea, and a Champions League loss to Galatasaray.
Their poor run comes despite splashing out 446 million pounds ($600 million) in the close-season transfer window and Carragher said manager Arne Slot should build the team around Alexander Isak and Florian Wirtz.
"It has to be Isak and Wirtz because they've spent the money on them and their age profile compared to Salah," Carragher said.
Liverpool are fourth in the league on 15 points from eight matches, four behind leaders Arsenal.
