Costa Rica's surprise victory against Italy on Friday saw the Three Lions eliminated from the World Cup at the group-stage juncture for the first time in 56 years.
It has seen Hodgson's position come under scrutiny, although FA chairman Greg Dyke announced his job was safe even before the Three Lions' fate was sealed.
He says the governing body want the 66-year-old to see out the his contract until 2016 and so too do the players, according to goalkeeper Hart.
"Me personally, I speak for the players when I say we are proud to play for Roy Hodgson," he said.
"He is a great manager, a very passionate man and someone I have an awful lot of respect for and I hope to continue playing for him."
Despite their early exit, Hart believes the Three Lions have made progress under Hodgson - something he knows "sounds like a stupid thing to say" given they will face Costa Rica on Tuesday with their fate already sealed.
England certainly looked a more attacking threat in Brazil compared to the European Championships in Ukraine and Poland, although defensively they appear to have regressed.
"We haven't scored enough goals and let too many in," Hart said.
"We are going to dissect but we have committed our lives to this tournament, thrown everything we can at it.
"We will continue to do that until the end of the group stages and whatever happens, happens. We have just not had the main stat, which is results.
"I've not had a great deal to do, I've not had a great deal to do in the tournament. I am man enough to say that.
"We are at the World Cup with quality strikers who given half a chance take it."
This has been a tournament of change for England, with only six of the squad from four years ago and 11 of the Euro 2012 party in this 23.
It is the second-youngest Three Lions squad ever taken to the World Cup, behind the side of 1958 - the last time they failed to make it out of their group.
Put to Hart that perhaps this squad is just not good enough, he replied: "I imagine some people might say that but we don't feel that inside the camp.
"Like I say, we have bags of quality, bags of people coming through.
"We have great leaders, great experienced players in that dressing room and have just come up short in those two games.
"You can't do that in a World Cup, you can't come up short in two games."
They are defeats Hart was quick to apologise for, both to the supporters at home and those who travelled to South America.
"I can't thank people enough for making this trip to Brazil, for making the efforts at home, for staying up for us, for revolving their work around us," the Manchester City goalkeeper added.
"If I wasn't here myself I would be doing the same. I can speak on behalf of the lads that we've not left anything to chance, we have tried everything."
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