Everton suffered another dismal outing on Saturday afternoon, squandering a 1-0 lead at Burnley to eventually lose 2-1 at Turf Moor, conceding two soft goals in their latest away defeat.
It's now six successive defeats on the road in all competitions for Everton, and with just two wins in 2018, manager Allardyce is already coming under increasing pressure.
'Big Sam' was only appointed as Ronald Koeman's replacement back in November, but disappointing results and an underwhelming brand of football has left fans calling for him to be sacked already.
Allardyce's contract runs until the end of the 2018-19 season, meaning Everton fans may yet have more than a year of Allardyce's reign still to go, but majority shareholder Farhad Moshiri has a decision to make this summer.
Carragher noted that Allardyce isn't showing that he can be anything more than a 'firefighter' at the bottom of the Premier League table, and Everton need more than just survival.
Carragher added that Allardyce comparing Everton to his former clubs Newcastle and West Ham hasn't been received well, and criticised his negative substitutions, as Toffees fans will always want to see their side go for wins.
“It reminds me of Roy Hodgson at Liverpool - it's just not the right fit,” said Carragher. “I think there were two jobs for Sam Allardyce: keep Everton up - they were five points above relegation when he came in - and to show he's more than just a firefighter. Can he be Everton's manager next season and show he's more than just keeping clubs up? At the moment, he's not showing that. The supporters didn't want him as they saw Allardyce as the manager at Sunderland or Crystal Palace - teams fighting down the bottom. Everton see themselves differently, challenging for European places.”
"The reason they're not happy with him is some of the things Allardyce says after games, comparing Everton to Newcastle and West Ham hasn't gone down well. When you are 0-0 away at Watford and you bring Morgan Schneiderlin on, Everton fans at least want to see their team try and win the game. That's been the criticism,” he added.