England entered the Brazil showpiece with high hopes but their new crop of young players failed to deliver, with consecutive 2-1 defeats to Italy and Uruguay ending their bid after just six days.
Hodgson's men finished the tournament with a disappointing goalless draw against Costa Rica on Tuesday and as a result, the former Liverpool and Fulham coach has been under pressure.
The 66-year-old was asked again about his future when told Italy manager Cesare Prandelli had offered his resignation following their elimination.
But Hodgson snapped back and said: "I told you (press) the other day (I won't resign) and I've nothing more to add.
"The FA (Football Association) 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 them to Euro 2016 and try to get some good results."
Hodgson was also pressed on the future of experienced midfielders Steven Gerrard and Frank Lampard.
And he does not want either player to retire from international football – as is widely expected – so he can involve them in the nation's UEFA Euro 2016 bid.
"It would be nice if they don't formally retire because when players do that they disappear from England selection," he added.
"(We 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."
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