Carroll has not played for his country since 2012 but has enjoyed a resurgence at the Hammers in recent weeks after returning from an injury lay-off.
The 27-year-old has scored six goals in his last eight appearances for the club, including a hat-trick against London rivals Arsenal a month ago, and appears to have returned to the kind of form which persuaded the club to pay £15 million for him three years ago.
Carroll has not had a look-in on the international picture over the last few years, but news of Danny Welbeck's injury, which is all but certain to rule him out of the competition, may have increased his hopes of earning a recall.
Gold insists that his forward must go to this summer's tournament in France, and says similarities between the country's other top strikers means there is no alternative but for Carroll to be selected.
He said, as quoted by the Daily Mail: "They don't have an alternative. The centre-forwards they've got are all very similar.
"All good players, don't misunderstand me, but if suddenly you think, do you know, I'm a manager, this isn't going right, I'd like to change this, I'd like to change tactics because maybe we need an aerial threat, who do you bring on?
"Another guy, 5ft 2ins? I don't think so. He's fit, strong and playing well."
With Welbeck injured, Harry Kane and Jamie Vardy both certainties after their superb seasons, and captain Wayne Rooney unlikely to be left behind regardless of whether he if fully fit or not, Hodgson has a big decision to make on the one or two other strikers he will include when he announces his squad on Monday.
Daniel Sturridge has struggled with injury this season but still has 12 goals from his 23 appearances, 33-year-old Jermain Defoe has netted 18 in 33 for a poor Sunderland side, and there could even be an outside shout for someone like teenage Manchester United prodigy Marcus Rashford.
However, Gold's point that all of those players are of a relatively similar ilk is understandable - none of them dominate in the air as a target man in the way that Carroll does - and the West Ham man would feel hard done by not to at least be seriously considered.