“President Trump’s anti-immigrant rants and policies are diametrically opposed to everything the Statue of Liberty stands for,” said Dani Lever, a spokeswoman for Mr. Cuomo. “His attendance would be truly ironic, and the mere symbolism of it should be offensive to all Americans whose ancestors were guided into this country by her torch.”

But some Republicans called the setting ideal for Mr. Trump to hold forth on immigration.

“The president should be welcome to come and deliver a speech at the statue, to highlight the benefits of legal immigration that built this country from its foundations,” said City Councilman Joseph Borelli, a Republican from Staten Island. “I suppose his base would enjoy him delivering an immigration speech while Governor Cuomo and Murphy sit silent.”

A White House spokesman did not respond to a request for comment.

Some politicos made sport of the whole thing.

The president could speak near the statue “as long as they cover her eyes,” quipped Bob Liff, a Democratic political consultant, who added, “Are they going to redact sections of Emma Lazarus’s poem?”

Bill O’Reilly — a Republican political consultant, not the former Fox commentator — said: “This could be the greatest episode of ‘Survivor’ ever — Chuck Schumer, Donald Trump and Jerry Nadler on an island together right in New York Harbor. Huge ratings.”

Politics aside, the museum should be pretty nifty: a 26,000-square-foot space that will feature the original glowing torch, interactive galleries and an immersive theater.

Which L.G.B.T.Q. figures should get monuments?