Amid reports of a plan to replace the former oil man with CIA director Mike Pompeo -- who is seen as more amenable to the president's hardline views -- Trump passed up an opportunity to publicly back his embattled top diplomat.
Asked if he wants Tillerson to remain in his post, Trump said only that "Rex is here" at the White House, a break from the usual expression of confidence in such a circumstances.
A senior White House official did not deny reports that Tillerson would be replaced, a rumor that has been circulating for months, but quibbled with the suggestion a plan was being rolled out.
The White House did little to scotch the rumors.
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"There are no personnel announcements at this time," said press secretary Sarah Sanders.
"Secretary Tillerson continues to lead the State Department and the entire cabinet is focused on completing this incredibly successful first year of president Trump's administration."
The State Department said that Tillerson enjoys his job as Washington's top diplomat and intends to stay on.
Trump and Tillerson have aired striking differences in public.
Trump surprised many observers when he tweeted that Tillerson was "wasting his time" pursuing contacts with North Korea, and Tillerson was quoted as having said that the president was a "moron."
The former ExxonMobil executive has also defended the Iran nuclear deal, which Trump last month disavowed.
Defense Secretary Jim Mattis, who has aligned himself with Tillerson's efforts in the North Korean crisis, said when asked about reports the top diplomat may be on the way out: "I make nothing of it, there's nothing to it."
It was not clear yesterday whether the president had given final approval to the reported cabinet changes, but he has previously tried to push advisors out rather than sack them outright.
Under the purported plan, Pompeo would be replaced at the CIA by Senator Tom Cotton, a Republican from Arkansas and a hawk on Iran and an important Trump ally on national security issues.
His departure would leave Republicans with another seat to defend, at a time when their Senate majority already in question.
Cotton won his seat easily in 2014, and is tipped as a future presidential contender, but Trump's unpopularity means Republicans can no longer assume victory in even deep red states.
The CIA declined to comment.
Analysts saw the reported move as evidence that Trump is trying to take more control over a foreign policy that at times has seem detached from the president's own rhetoric.
"Pompeo and Cotton have both convinced Trump they fit the bill," he added. "Overall, it's a sign that Trump wants more control over his own foreign policy.