On Thursday, news outlets quoted senior administration officials as saying the president could name Central Intelligence Agency Director Mike Pompeo to replace Tillerson within the next several weeks. Both the White House and the State Department denied the reports.
Earlier Friday, as Tillerson stood before reporters alongside Libyan Prime Minister Fayez al-Sarraj, he was asked about reports that the White House wanted him to resign.
Tillerson responded curtly, saying, "It's laughable. It's laughable."
Questions about the possible Cabinet shake-up dominated Thursday's State Department briefing, with spokeswoman Heather Nauert saying Tillerson was "unflappable" and going about his business, which includes a planned trip to Europe next week to meet with NATO leaders. But she did repeat several times that the "secretary serves at the pleasure of the president."
On Friday, Tillerson met with Libya's al-Sarraj at the State Department before having lunch with Trump and Defense Secretary Jim Mattis at the White House later in the day.
Despite Trump's denial on Friday that he wanted to fire Tillerson, the Wilson Center's Aaron David Miller said reports of a replacement pick might be a not-so-subtle attempt to send the secretary of state a message that his services are no longer needed, since the president has demonstrated his dislike of firing people.
"So it is possible that he [Trump] is laying the groundwork for what they would like to achieve, which is a kind of 'immaculate resignation,' " said Miller, a senior adviser to several Republican and Democratic secretaries of state.
Miller said foreign leaders are likely more concerned about Tillerson's perceived "fraught" relationship with Trump than reports that his days as chief U.S. diplomat are numbered.
"I think our allies and adversaries understand that Tillerson does not necessarily speak for the president," Miller told VOA. "And that is the worst possible position for a secretary of state to be in, because it takes our allies and adversaries five seconds to figure that out. And if that is in fact the case, then the credibility, the relevance, the resonance of the secretary of state is profoundly undermined."
Ronald Neumann, a former U.S. ambassador to Afghanistan, Bahrain and Algeria, said rumors that Tillerson's job was not secure would affect the secretary slightly as he carried out his diplomatic duty, but he noted Trump had already left some foreign officials with questions about U.S. foreign policy.
"The greatest effect on his ability to do his job is whether people are confident that he speaks for the president, and it is the president who makes that uncertain from time to time with tweets and position changes," Neumann said.
The former ambassador said Tillerson had not had an easy tenure.
"Secretary Tillerson has been under more public attack in the American press than I have ever seen a secretary under for such a long period. And that is bound to be known by his NATO colleagues," said Neumann, president of the American Academy of Diplomacy in Washington.
He also noted that Tillerson's management of the State Department had triggered more criticism than his policy positions.
"It is important to note that most of the attack has been about his internal management of the department, not about the foreign policy positions he has been taking, where there is a back and forth with the White House," Neumann said. "But I think, by and large, NATO members are reasonably comfortable with what he has been doing with diplomacy."
Tillerson will travel to Belgium, Austria and France December 4-8 and attend a NATO foreign ministers meeting. The State Department said his last stop would be in Paris, where he will meet with senior French leaders to discuss cooperation on Syria, Iran, Lebanon, Libya, North Korea and the Sahel.
Source: Voice of America