"We're far from being favourites but you've got to shy away from 'impossible' in football," Hodgson told the BBC.
Hodgson, appointed on May 1 on a four-year contract, used Chelsea's Champions League win in the same month as an example of how unpredictable football was.
"There are plenty of events in football that you could have said were impossible but people have done it.
"It wasn't likely that Chelsea would become champions of Europe, especially having to change their manager mid-stream but they did it. That was by beating Barcelona and Bayern Munich - two of the best teams in Europe, so I'd like to keep that dream alive."
England reached the quarter-finals of the European Championship this year but lost on penalties to Italy and have failed to get past the last eight of a major tournament since Euro 1996, when they made the semi-finals on home soil.
The 1966 World Cup winners have a battle on their hands to make it to Brazil as they are second in Group H, two points behind Montenegro after four games and with six fixtures remaining in 2013.
Hodgson said that, given the amount of home-grown talent, he felt optimistic about the future and hoped that with so many players of a high standard there would be increased competition for places.
"We have a lot of quality players - you could reel 30-40 players off the top of your head quite easily and add some of these young players who seem to have a very bright future in front of them," he said.
"I think the main problems are whether the main figures are available at the right time when you need them. Do we have enough competition in the really important positions in the teams?"
The influx of foreign players into the Premier League had affected the careers of young English players in recent years but the current Manchester United team showed that home-grown talent was leading the way, Hodgson said.
"I know there are a lot of foreigners in the league but there are plenty of English ones. There were seven English players playing for Manchester United the other day and they're top of the Premier League."
