Mignolet became Liverpool's first choice last season following his arrival from Sunderland in a deal worth £9million, though he has failed to fully convince fans at Anfield.
The Belgium international has come under scrutiny this term, having already conceded 12 goals in eight matches in all competitions.
Liverpool have been linked with a move for former Barcelona keeper and free agent Victor Valdes.
Carragher told Sky Sports on Monday that Mignolet does not make enough "defining" saves after he was unable to deny Everton captain Phil Jagielka from equalising in sensational fashion on Saturday as Liverpool go in search of Premier League glory.
"I think the best goalkeepers who win big trophies make big saves at big moments," said Carragher.
"That was a big moment [the Jagielka goal]. It's not a massive mistake, but can he do better?
"Everyone always talks about Steven Gerrard slipping and costing Liverpool the league and you can never look at one individual and one mistake, but Demba Ba still had to go through.
"You're looking at your keeper and thinking: 'Come on, win us the league. Make that save that's going to be a defining moment in the season.'"
Carragher added: "We can all talk about goalscorers, players who win the league and make a big difference, but goalkeepers make a massive difference and at those big moments you've got to make big saves.
"Up until now in his Liverpool career he hasn't made those big saves. If it continues I can see Liverpool looking for another goalkeeper.
"There was talk of Victor Valdes because he is a free agent, but he has got to improve."
