Hodgson acknowledged that Carroll's aerial threat would be an asset at Euro 2016, but insists that he will not pick a player solely because they are the flavour of the month.
"We know Andy, we know his qualities. In an ideal world, certainly, if you had a lot of places at your disposal you'd always include a guy like him because he is exceptional in the air. But whether or not I'd be able to include him with all these others, I don't know," he told reporters.
"But it's not out of the question because he's English, he's a centre-forward, he's scoring, he's a good player, so who knows? When the time comes and I'm looking at what's available, maybe he'll be there. But I'm certainly not going to pick him on the basis of he scored a hat-trick so therefore he has to be in the England team.
"I think that's poor work from a mass media point of view that every time someone scores a goal and he's English 'why is he not playing for England?' Unfortunately, the answer often is they're not playing for England at this particular moment because there are plenty of others who have done bloody well. And also I'd ask you not to forget we have had the best qualifying campaign I think England has ever had, with 10 straight wins.
"We've only lost [two] friendlies of the last six or seven and we have played teams like Spain, France, Holland, Germany and Italy, so we haven't exactly been playing the Liechtensteins of the world. Maybe before we start wanting to throw out the guys who have done so well for the latest shout, the latest ice-cream flavour, we should show a little bit of respect to what they have done as well."
England begin their Euro 2016 campaign against Russia on June 11 before taking on Wales and Slovakia in their other two group games.