Late in the day, the White House tried to douse the furor. “In response to media speculation and related questions being posed to the administration, the White House yet again confirms that the president is not considering or discussing the firing of the special counsel, Robert Mueller,” Ty Cobb, a White House lawyer, said in a statement.

The president’s tweet followed a statement by Mr. Trump’s personal lawyer, John Dowd, on Saturday calling on the Justice Department to end the special counsel investigation. Mr. Trump followed up that evening with a tweet arguing that “the Mueller probe should never have been started in that there was no collusion and there was no crime.”

The two weekend tweets were the first time Mr. Trump has used Mr. Mueller’s name on Twitter, not counting a message he once retweeted, and reflected what advisers called a growing impatience fueled by anger that the investigation was now looking at his business activities.

The New York Times reported last week that Mr. Mueller has subpoenaed records from the Trump Organization. Mr. Trump’s lawyers met with Mr. Mueller’s team last week and received more details about how the special counsel is approaching the investigation, including the scope of his interest in the Trump Organization.

For months, Mr. Trump had been reassured by his lawyers that the investigation would wrap up soon — by Thanksgiving, then Christmas, then New Year’s. But with the expansion into Mr. Trump’s business, it seems increasingly clear that Mr. Mueller is not ready to conclude his inquiry.

A top adviser to Mr. Trump said on Sunday that the White House had grown weary of the inquiry. “We have cooperated in every single way, every single paper they’ve asked for, every single interview,” Marc Short, the president’s legislative director, said on “Face the Nation” on CBS. “There’s a growing frustration that after a year and millions and millions of dollars spent on this, there remains no evidence of collusion with Russia.”

A president cannot directly fire a special counsel but can order his attorney general to do so. Even then, a cause has to be cited, like conflict of interest. Since Attorney General Jeff Sessions, a former campaign adviser, has recused himself from the Russia investigation — to Mr. Trump’s continuing irritation — the task would fall to the deputy attorney general, Rod J. Rosenstein.