The hope from the White House, the source said, is to push out the plan to replace Tillerson and then "wait for him to punch out."
The news that the White House is seriously considering replacing Tillerson with CIA Director Mike Pompeo comes as Trump remains deeply frustrated with his secretary of state, another source familiar with the President's thinking said.
And the plan is not just being considered at the staff level, but by the President himself, the source said.
The issue could come up during a scheduled lunch between Trump, Tillerson and Defense Secretary James Mattis at the White House Friday.
The attempt at public shaming is just the latest such instance emanating from the Trump White House, where the President has repeatedly publicly undermined his secretary of state and publicly berated his attorney general.
His public response to the reports on Thursday signaled no desire to spare his secretary of state -- who still has not denied having called the President a "moron" in private.
"He's here. Rex is here," Trump offered, noting that Tillerson was in the building -- but not in the room -- at the time.
The response was equally uninspired from the White House briefing room, where White House press secretary Sarah Sanders said simply that "when the President loses confidence in someone, they will no longer serve in the capacity that they're in."
The timing of Tillerson's expected departure remains uncertain, but multiple officials said the shake-up could come at the end of the year or early next.
"The clock is ticking," the source familiar with the White House's latest thinking said.
Mattis said Thursday that "there's nothing to" the rumors about Tillerson being forced out by the White House.
"I make nothing of it, there's nothing to it," Mattis said before a bilateral meeting with Libya's Prime Minister.
But the White House's tentative plans to replace Pompeo at the CIA with Sen. Tom Cotton, R-Arkansas, could present a wrinkle in the strategy.
Cotton isn't up for reelection until 2020, and moving him to the CIA role would put another Senate seat in play in 2018 -- at a time when Republicans have a razor-thin majority. That is giving West Wing officials pause, a source close to the White House said.
If Cotton were to take the CIA job, Arkansas Gov. Asa Hutchinson, a Republican, would select someone to fill the vacant seat -- and Republican officials would have a tight time frame to field a candidate who can win statewide in 2018, even though Arkansas is a deep red state.
The GOP would have to field a candidate by March 1, when the filing period ends, and the primaries are currently scheduled for May 22.
But the process could hit a roadblock as the governor's Senate appointee could not run in 2018, according to a clause in the state's constitution. But the Arkansas Secretary of State's office raised questions about that clause, noting that US Constitution may offer a conflicting view that could override the state statute.