The top executive of the Smithsonian, Lonnie G. Bunch III, was summoned to a White House lunch on Thursday in the midst of President Trump’s efforts to review and change content at the institution’s museums.

Neither side publicly discussed what happened at the meeting, which came as the White House increasingly pushes for a role in deciding what is presented in Smithsonian exhibits. But a White House official described the lunch as “productive and cordial.”

Mr. Bunch, the secretary of the Smithsonian, and Mr. Trump were joined by Lindsey Halligan, a special assistant to the president who is leading the White House review.

Mr. Trump has said the Smithsonian’s museums present “divisive narratives” and “race-centered ideology,” and this month the administration sent Mr. Bunch a letter announcing a wide-ranging review of current and planned Smithsonian exhibitions. The examination would involve scouring wall text, websites and social media at eight of the institution’s 21 museums “to assess tone, historical framing and alignment with American ideals.”

The White House said then that museums would be required to adjust any content that the administration finds problematic within 120 days, “replacing divisive or ideologically driven language with unifying, historically accurate and constructive descriptions.”

Since then, the Smithsonian leadership had been debating how to respond to the White House order in a way that reasserted its independence without escalating tensions with the administration. The institution had announced its own review of content several weeks ago as part of an effort to eliminate any content that was inappropriately biased or partisan.