Costa Rica's 1-0 win over Italy on Friday condemned the Three Lions, who have lost their opening two games at the finals for the first time, to elimination even before they have played all three of their Group D matches in Brazil.
Football Association chairman Greg Dyke has stressed that this does not change plans made when Hodgson signed a four-year contract in 2012, and while Carragher expects the 66-year-old to remain in charge and take England to the 2016 European Championships, he has warned that his position may become "untenable" should his side lose their last group game to Costa Rica in Belo Horizonte on Tuesday.
"I think it is [right that he stay on]," he told Sky Sports News. "But I think the last game is also important. He 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. Hopefully the youngsters will show what they can do in that game and in qualifying for Euro 2016, which I'm sure we'll walk through.
"But it is right Roy Hodgson is staying and I'm glad Greg Dyke has come out and said that. There would have been a lot of talk and hopefully this will kill that talk."
Ex-Liverpool defender Carragher earned 38 England caps between 1999 and 2010, representing them at two World Cups.