Hodgson has been reassured of his future as national boss by the FA, despite being knocked out of this summer's World Cup in Brazil after just two games.
Carragher believes that the governing body are right to keep the 66-year-old on, but warned that defeat to Costa Rica could change the complexity of the situation completely.
The Costa Ricans currently sit top of Group D after beating both Uruguay and Italy, with Bryan Ruiz's winner against the Italians confirming England's departure from South America.
"I think the last game is also important," the former Liverpool defender is quoted as saying by Sky Sports.
"He [Hodgson] is obviously going to keep his job but you certainly don’t want to be going home with three defeats and Costa Rica look a decent side now.
“It’s OK Greg Dyke saying that but, if England suffer another defeat, it may become untenable then."
Meanwhile, Three Lions goalkeeper Joe Hart has offered the national coach his backing, praising him for his "passion" for the job.
"Me personally, I speak for the players when I say we are proud to play for Roy Hodgson," Hart said.
"He is a great manager, a very passionate man, and someone I have an awful lot of respect for. I hope to continue playing for him."
If Carragher's suggestion bares any fruit, however, Hodgson is in for a make-or-break 90 minutes against Costa Rica, with the Central American side high on confidence after back-to-back wins.
Coach Jorge Luis Pinto has already expressed his desire to beat England despite already sealing qualification to the knockout stages, while Ruiz and Joel Campbell have both revelled in having a point to prove to their Premier League clubs - Fulham and Arsenal respectively.