Hodgson and his 23 players touched down in England yesterday after their worst World Cup campaign in history.
England finished bottom of Group D with just one point.
Hodgson refused to walk out on England after their early exit was confirmed last week and the 66-year-old has had no change of heart despite the dull draw against Costa Rica.
Hodgson snapped back at one reporter when it was pointed out that Italy manager Cesare Prandelli had offered to resign following his team’s elimination from the tournament on Tuesday.
“I told you the other day (he will not resign) and I’ve nothing more to add,” the England manager said before adding: “The FA have asked me to continue. They want me to continue. I’m very happy that they want me to continue.
“I think there is an interesting group of players here to work with. And I get no feeling whatsoever that any of them will want me to resign.
“I have no reason to do so. I have no intention to do so.
“I don’t want to be compared to other people.
“The reason I’m staying on is that I’m not a quitter, I believe in this team and that the FA seriously want me to keep doing this job, as do the players, so therefore I will continue to do it and I’ll try and lead the team to Euro 2016 and try to get some good results.”
Hodgson will, over the next few weeks, hold discussions with members of Club England – the body that oversees the day-to-day running of the national side.
It is understood that there will be no requirement for Hodgson to submit a full report to the organisation regarding England’s World Cup failure, though.
When the “grieving process”, as Hodgson put it, is over, the England manager will sit down and analyse videos of the team’s failings in Brazil.
“Our video man will be cutting that up now although he won’t have a chance to show me that for a while,” Hodgson said.
“But certainly we will all sit down together before we start our next campaign to see what did we learn from the World Cup?
“What did we learn about our players? What did we learn about the way we want to play?
“What do we think we can push forward and if we made any glaring mistakes then what can we do to make sure we don’t make them again?”
Video analysts formed part of the vast army of staff at Hodgson’s disposal in South America.
A nutritionist, a chef, fitness trainers and sports scientists were recruited to help England this summer, as well as sports psychiatrist Dr Steve Peters.
Hodgson will consider asking Peters, renowned for his work with Olympic athletes, to help England at Euro 2016, should they qualify.
“Yes. He’s been very, very good,” Hodgson said when asked if he would consider taking a psychiatrist to Euro 2016.
“He hasn’t played a major part in terms of global team talks, but what he has done is he’s been very good with the individual players and the ones he has actually worked with.
“Others have bought into it as well so I think going forward, the nutritionist, the extra fitness coach we took from Manchester United and Steve Peters have all been very valuable members of our squad and they helped us prepare better.”
Steven Gerrard and Frank Lampard are two who may not be available to Hodgson should England make it to France.
The two midfielders have been part of the England set-up for the last 15 years, but with a combined age of 70, the 0-0 draw against Costa Rica may be their last in England shirts.
Hodgson wants them to continue, although he appears open to the idea of resting them for certain matches.
He said: “It would be nice if they don’t formally retire because when players do that they disappear from England selection.
“(England could) leave it to a dialogue perhaps whereby at any moment in time I could speak to one of them and say ‘look, do you want to play in this game, are you up for it?’. That’s how I’d really like to see it.”
If Gerrard does retire, then Hodgson will have to select a new captain.
It is understood that Hodgson considers Wayne Rooney to be a serious contender to take over the armband.