The Uruguayan is heavily tipped to move to Premier League rivals Arsenal after the 26-year-old rejected the chance to apologise to manager Brendan Rodgers following recent media interviews hitting out at the club's hierarchy.
Reds owner John W Henry recently reiterated his stance that Suarez is not for sale, and Carragher feels that Liverpool would look foolish if they allowed the player to leave during the current transfer window.
"I expect him to stay at Liverpool now," Carragher told the Daily Telegraph. "I think if you come out as owner and say that, and then something happens, you make yourself look stupid.
"I'm glad he has done it and I think everyone in football should be glad he has done it. It is maybe a message for players.
"I trained with him once this summer and you are talking about probably one of the best trainers I have ever seen. I've only seen him once, God knows what he has been like all pre-season, but he hasn't been his normal self."
Suarez was Liverpool's top scorer last season and finished with 23 league goals.