There have been calls for Carroll, 27, to make an international return after his recent hat-trick against Arsenal.
But Hodgson said: "Every time someone scores a goal and he's English it is 'why is he not playing for England?'
"I'm certainly not going to pick him on the basis that he scored a hat-trick so therefore has to be in the England team."
Ex-Newcastle and Liverpool forward Carroll has scored two goals in nine England appearances, but has not featured for the national side since October 2012.
Tottenham's Harry Kane, the Premier League's top-scorer with 22 goals this season, and Leicester striker Jamie Vardy, who has netted 21 times, have established themselves in the England set-up and both played in last month's friendlies against Germany and the Netherlands.
Arsenal forwards Danny Welbeck and Theo Walcott and Liverpool striker Daniel Sturridge were also in those squads, while captain Wayne Rooney has now returned to the Manchester United side following injury.
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Carroll has scored seven league goals this season and Hodgson will name his 23-man squad for this summer's tournament in France on 12 May.
"We know Andy, we know his qualities," added the 68-year-old.
"In an ideal world, 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."
The England boss hinted that even skipper Rooney might not be assured of a place after his two-month lay-off with a knee problem.
"There's no disguising the fact that people like (Arsenal midfielder) Jack Wilshere, (Manchester City forward) Raheem Sterling, (Liverpool midfielder) Jordan Henderson, (Arsenal midfielder) Alex Oxlade-Chamberlain, Wayne Rooney - a year ago they were five players who were getting into every squad," said Hodgson.
"Things have happened to them and we have to reassess.
"If we can be sure they are at the level of fitness and capability to produce the performances they produced a year ago they will be serious contenders. If we are not convinced, we have to give it some thought."

Mike Peters, lead singer of The Alarm, is encouraging US congressmen and senators to 'Get On The List' on Capitol Hill on Tuesday.
Three-time cancer survivor Mr Peters co-founded the Love Hope Strength Foundation in 2007 to encourage more people to sign up as potential lifesavers.
The US visit follows a similar recruitment drive in Westminster.
Ahead of the event, Mr Peters encouraged people in the USA to write to their local representative asking them to attend the donor drive at Capitol Visitor Centre.
Potential donors fill out a consent form and have their cheeks swabbed. Their details are then stored anonymously on an international bone marrow registry until they are 61-years-old.
It is hoped the event will grow the number of US donors and improve the chances of finding matches for patients suffering from a blood borne cancer or illness.
Mr Peters said the foundation has been campaigning for this day for many years.
"It is so exciting to feel that all the hard work and effort, made possible with the support of DKMS - We Delete Blood Cancer, and so many others, is finally coming to fruition," he said.
"People are realising that one day it could be someone that they know who needs a bone marrow donor, and the more people that are on the list, the better everyone's chances of finding an unrelated donor area."
The event in the House of Commons in 2013 registered more than 100 MPs and staff members, and MP John Glen became a lifesaving match a few months later.

The Fire and Rescue Service said they received a call just before 08:30 GMT on Thursday.
The man was found dead in an upstairs bedroom, after fire crews arrived at the house in Drumgay Close, in the Kilmacormick area. They wore breathing apparatus and used thermal imaging cameras to locate the occupant.
The Fire and Rescue Service are working to establish the cause of the fire.
England manager Roy Hodgson has played down West Ham striker Andy Carroll's chance of being selected for Euro 2016.
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A Welsh rock star has taken his bone marrow donor drive to Washington DC.
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A man has been found dead after a house fire in Enniskillen, County Fermanagh.