WASHINGTON — Aboard Air Force One on a flight home from Europe last July, President Trump and his advisers raced to cobble together a news release about a mysterious meeting at Trump Tower the previous summer between Russians and top Trump campaign officials. Rather than acknowledge the meeting’s intended purpose — to obtain political dirt about Hillary Clinton from the Russian government — the statement instead described the meeting as being about an obscure Russian adoption policy.

The statement, released in response to questions from The New York Times about the meeting, has become a focus of the inquiry by Robert S. Mueller III, the special counsel investigating Russian interference in the 2016 election. Prosecutors working for Mr. Mueller in recent months have questioned numerous White House officials about how the release came together — and about how directly Mr. Trump oversaw the process. Mr. Mueller’s team recently notified Mr. Trump’s lawyers that the Air Force One statement is one of about a dozen subjects that prosecutors want to discuss in a face-to-face interview of Mr. Trump that is still being negotiated.

The revelation of the meeting was striking: It placed the president’s son and his top campaign officials in direct contact with a Russian lawyer who promised damaging information on Mrs. Clinton, and an email to the president’s son emerged saying that the information was part of Russia’s effort to help the Trump campaign. The special counsel is investigating how those revelations were handled in real time in part because the president was involved in his administration’s response.

Some lawyers and witnesses who have sat in or been briefed on the interviews have puzzled over Mr. Mueller’s interest in the episode. Lying to federal investigators is a crime; lying to the news media is not. For that reason, some of Mr. Trump’s advisers argue that Mr. Mueller has no grounds to ask the president about the statement and say he should refuse to discuss it.