Hart and his team-mates will be flying home next week, Costa Rica’s 1-0 win over Italy having consigned them to a Group stage exit for the first time in 56 years.
England will be playing for pride against the surprising Costa Ricans on Tuesday following 2-1 losses to Italy and Uruguay but FA chairman Greg Dyke has already said that Hodgson will not pay for the disappointing campaign with his job.
And Hart says the squad will welcome the news that Hodgson will be masterminding their Euro 2016 campaign.
"Me personally, I speak for the players when I say we are proud to play for Roy Hodgson," said the Manchester City man.
"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."
Hart thanked the fans for their support and said they could rest assured that the squad had given it their all.
"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," he 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.
"We haven't scored enough goals and let too many in. We are going to dissect it but we have committed our lives to this tournament, thrown everything we can at it.
"We have just not had the main stat, which is results.
"I've not had a great deal to do in the tournament, I am man enough to say that.
"But we are at the World Cup with quality strikers who, given half a chance, take it."
Asked if he thought the squad was not good enough. Hart said: “I imagine some people might say that but we don't feel that inside the camp.
"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."