A camera pans over a photograph of Mount Rushmore. The stone sculpture shows the faces of George Washington, Thomas Jefferson, Theodore Roosevelt and Abraham Lincoln. At the end, an illustration reveals an imagined new face — President Trump.

Mount Rushmore has featured four iconic American presidents since the massive sculpture was completed in 1941: George Washington, Thomas Jefferson, Theodore Roosevelt and Abraham Lincoln.

Photo illustration by The New York Times

President Trump has made no secret of his fondness for Mount Rushmore and his desire to join its rock-star lineup.

During his first term, Mr. Trump told Kristi Noem — then a U.S. representative from South Dakota, now Mr. Trump’s Secretary of Homeland Security — that his “dream” was to be on Mount Rushmore. She later gave Mr. Trump a model of Mount Rushmore with his face on it.

The idea has resurfaced since Mr. Trump returned to office. A congresswoman from Florida sponsored a bill in January to “direct the Secretary of the Interior to arrange for the carving of the figure of President Donald J. Trump on Mount Rushmore National Memorial.” It was referred to the House Committee on Natural Resources, which has yet to act on it.

In March, Interior Secretary Doug Burgum said in an interview with Lara Trump, Mr. Trump’s daughter-in-law, that “they definitely have room” for Mr. Trump’s face on Mount Rushmore.

Wait. Is this possible?

As with all things Trump, it can be hard to decipher the difference between everyday rhetoric and future action. But those in charge of the memorial are taking such overtures seriously.

Mount Rushmore, in the Black Hills of South Dakota, in 1967.

Charles E. Rotkin/Corbis, via Getty Images

The National Park Service, which oversees Mount Rushmore National Memorial, and which is currently led by Mr. Burgum, has cited two reasons that more faces cannot be added. First, it considers Mount Rushmore to be a completed work of art. Second, there is no room. “The carved portion of Mount Rushmore has been thoroughly evaluated, and there are no viable locations left for additional carvings,” the park service said in a statement.

Those are the frames for the debate, now and in the future. One is philosophical. The other is geological. One is a question of should; the other is a question of could.

Mount Rushmore is a cultural touchstone, American shorthand for the best of the best. “Who belongs on your Mount Rushmore?” is the opening to a debate on any niche subject, as if we might carve such a monument to athletes, musicians or writers.

But the real thing is about presidents. Gutzon Borglum, the sculptor who designed the massive carvings and oversaw their construction for 14 years, starting in 1927, chose the four presidents and their precise locations.

What would he make of President Trump? And where would he put him?

Mount Rushmore, 1927

1936-1937

1933-1934

1939

Mount Rushmore, 2025

A flyover animation of a 3-D model of Mount Rushmore shows the details of the carved mountain and the surrounding rock formations. Archival images of the construction of the national memorial are overlaid within the 3-D scene to show how it was created, including images of complex scaffolding and support structures to allow crews to work on the rock for more than a decade.

Mr. Borglum had quite an imagination. Historical photos show that Mount Rushmore, more bluff than peak, was a jumble of weathered rock formations that stood out among South Dakota’s Black Hills. The southeast face had promise and great light.