The line of questioning adds to indications that Mueller views false or misleading statements to the press or to the public as obstruction of justice. That could set up a potential flashpoint with the White House and the Trump legal team should that become part of any final report from the Mueller investigation.
Mueller hasn't addressed the issue publicly, but prosecutors have dropped hints that they view public statements as possibly key in influencing witnesses.
Court filings from the plea of Michael Cohen, the President's former personal lawyer, included allegations related to false public statements -- not usually considered illegal since they aren't made directly to investigators. A December sentencing memo filed by Mueller's office notes that Cohen's lies were amplified in public statements, "including to other potential witnesses." The memo said this was done partly "in the hopes of limiting the investigation into possible Russian interference in the 2016 US presidential election -- an issue of heightened national interest."
The President's legal team took notice of the Cohen plea documents and believe the special counsel is pursuing a thin legal theory when it comes to potentially criminalizing public statements, a source briefed on the matter said. They believe the President's comments are protected under the First Amendment and that there's a difference between a plea deal and the President's position, which is to fight the allegations.
"Read CNN article that (Mueller) wants to use the President's public comments as part of an obstruction claim. According to this latest oppressive legal theory you can't even defend yourself," Giuliani said.
Whenever Mueller's investigation concludes, the moment is expected to mark the beginning of a new political and legal fight over the probe's findings.
While Mueller is widely expected to not try to bring charges against the President, following long-standing Justice Department legal guidelines, investigators could try to weave into their report examples of how the President used public statements to try to obstruct the investigation. The Mueller team's questioning also shows their interest in obstruction extends beyond the President's firing of FBI Director James Comey
One witness who worked closely with Trump told CNN that, based on the questions asked, it is clear Mueller is looking at the President's "changing stories" as a way to possibly show corrupt intent in the obstruction probe.
Another episode centers on Trump's attempt to have his then-White House counsel Don McGahn disavow articles reporting that McGahn threatened to quit over Trump's pressure to fire Mueller, according to one source familiar with the matter. McGahn refused to publicly say that the stories were false, which prompted more anger from the President.
The President publicly has said he wouldn't fire Mueller and hadn't thought about it, which other witnesses have told Mueller's investigators isn't true.
Meanwhile, as clues mount that Mueller is ending his multi-year investigation in the coming weeks, new political and legal fights are emerging over who will actually be able to read his findings.
It's a battle the White House and Congress are already preparing for to determine if any of the report Mueller produces goes to Congress or any part of it becomes public.
President Donald Trump wouldn't indicate Thursday whether he wants Mueller's final report to be made public, saying to reporters, "We'll have to see."
But behind the scenes at the White House, an intense effort is underway to keep a lot of what Mueller submits private. The new White House counsel, Pat Cipollone, has been beefing up his staff to deal with this fight, hiring 17 new lawyers including three deputy counsels, a senior administration official said. More could be hired.
Mueller's investigators have been reviewing more than a million pages of documents that the President and the White House legal team voluntarily turned over without waiving privilege.
Trump's legal team and Cipollone believe that a large portion of the information in Mueller's investigation should be protected by executive privilege. The Trump lawyers envision a heavily redacted version of the report to perhaps become public.
For a year, the Trump lawyers and the Mueller team have held on-and-off discussions for a possible interview with Trump, though the President has been counseled not to talk. His lawyers have told the special counsel they will not agree to an in-person interview, the president's attorney Rudy Giuliani told CNN on Wednesday.
But the President agreed to answer a limited set of questions in writing regarding matters that occurred before the 2016 election. Those responses were submitted before Thanksgiving.
While Mueller indicated there were some follow-up questions, Trump's lawyers resisted and demanded to know which areas would need additional questions, Giuliani told CNN.
Another source familiar with the discussion said the special counsel has been told that the President won't answer any more questions.
The special counsel's office has not responded and there has been no discussion between the two sides since before Christmas, Giuliani said.
Even as the investigation continues, the Mueller team has already begun writing parts of the report laying out the investigation's findings, sources familiar with the matter said.
One person who knows Mueller and soon-to-depart Deputy Attorney General Rod Rosenstein said the expectation is that it will be an extremely detailed written analysis and finalized after the investigative work is concluded. This person said both Mueller and Rosenstein are proponents of a written work product, leading to the belief it will be a thorough report.
The Justice Department regulations governing the special counsel calls for Mueller to "provide the Attorney General with a confidential report explaining the prosecution or declination decisions reached by the Special Counsel" when his work is completed.
The special counsel's report is expected to answer a number questions that follow Rosenstein's order appointing him, including whether anyone in the Trump campaign illegally coordinated with Russians who interfered in the 2016 elections.
But investigators have also sought, through interviews and documents, to determine: whether Trump obstructed justice; whether Trump knew in advance about the Trump Tower meeting with Russians in June 2016; whether campaign contacts sought stolen Democratic Party emails; whether foreign money illegally came into the election; and what gave rise to the Republican Party changing its platform in 2016 on Russian intervention in Ukraine.
The regulations don't require Mueller to share his findings with the public.
If Mueller and the Justice Department do not make the report public -- and if the White House wants to keep it largely secret -- Congress could issue a subpoena to try to force its release.
Democrats, who now control the House of Representatives, have said they will fight to make sure the report is made public.
California Rep. Adam Schiff, the top Democrat on the House Intelligence Committee, told CNN recently he's prepared to do "everything possible so that the public has the advantage of as much of the information as it can."
"Now, there may be parts of the report that have to be redacted because they involved classified information or they involve grand jury material," Schiff said on CNN's "State of the Union with Jake Tapper" just before Christmas. "At the end of the day, this case is just too important to keep from the American people what it's really about."
One of the likely points of contention between Trump and congressional opponents is likely to center on whether the President obstructed justice with some of his actions during the investigation.
Trump's team refused to answer any obstruction-related questions. They've argued that this part of the probe goes to the heart of the President's constitutional responsibilities. That could make any effort by Democrats to focus on that issue more contentious. Trump's lawyers have even sought to resist answering questions about the transition period -- the period of time between the election and the inauguration -- arguing that it is part of the incoming administration and should also be covered by executive privilege.
But Schiff said that the administration has already set a precedent with its cooperation with Republicans in Congress investigating Trump's 2016 opponent Hillary Clinton.
"For the last two years, I have been warning the Justice Department, as they have been turning over tens of thousands, hundreds of thousands of pages of investigative files in the Clinton e-mail investigation, that whatever precedent they were going to set, they were going to have to live by," Schiff said.
CNN's Dana Bash contributed to this report.