At one point last summer, when a lull in talks had the president’s attorneys worried that Mueller was seriously contemplating a subpoena, White House lawyer Emmet Flood wrote a memo laying out the legal arguments for protecting the president’s executive privilege. He sent the document to Mueller’s office and to the deputy for top Justice Department official Rod J. Rosenstein, who oversaw the probe, according to two people familiar with Flood’s outreach.

Meanwhile, the Trump lawyers sent a steady stream of documents and witnesses to the special counsel, chipping away at Mueller’s justification for needing an interview with the president.

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Trump’s lawyers, citing the independent counsel investigation of Espy, argued that to justify a subpoena of Trump, Mueller needed to prove that he could not get the information in any way other than by asking the president.

“No matter what question they would say they wanted to ask, I felt confident we could turn it over and say, ‘You already have the answer to it,’ ” Giuliani said. “If they said, ‘Why did you fire Comey?’ I’d give them five interviews, and particularly the Lester Holt tape, where he goes into great detail as to his reasons.”

Giuliani was referring to Trump’s May 2017 interview with the “NBC Nightly News” anchor in which the president said he was thinking about “this Russia thing” when he fired James B. Comey as FBI director, one of the actions Mueller was investigating as possible obstruction of justice.

All the while, Giuliani said, the legal team was not convinced that it would have prevailed in court. “Honestly, I don’t know who would have won,” he said. “I think our argument got better as time went on. But I don’t know if we would have won.”

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In the absence of an interview, Trump’s attorneys offered Mueller a substitute: The president would provide answers to a set of questions about Russia and the campaign, submitted in writing. But, citing executive privilege, they refused to provide answers to questions pertaining to the president’s time in office — questions that went to the heart of the special counsel’s inquiry into possible obstruction of justice.

However, the process of compiling answers dragged. Trump’s lawyers found it difficult to get the president to focus on drafting the submission, according to people familiar with the sessions.

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Finally, in late November 2018, the lawyers sent Trump’s answers to Mueller.

In December, Mueller’s team made one more request for an interview with the president.

And in January, the special counsel’s office contacted Trump’s lawyers to ask some follow-up questions, according to people familiar with the request.

But Trump’s lawyers again declined.They neither agreed to an interview nor answered the additional questions.

Two months later, Mueller submitted his report without having spoken to the president. The investigation was over.

A federal judge on Thursday struck down a Trump administration rule that allows small businesses to band together and set up health insurance plans that skirt requirements of the Affordable Care Act.

The rule is “clearly an end-run around the A.C.A.,” said the judge, John D. Bates, of the Federal District Court for the District of Columbia.

The ruling was the second big defeat this week for President Trump on a top-priority item on his health care agenda as he has sought to use the courts to obliterate his predecessor’s signature achievement. Another judge on Wednesday blocked Medicaid work requirements in Arkansas and Kentucky.