By Jeremy Diamond, Jeff Zeleny and Manu Raju, CNN

(CNN) -- President Donald Trump's appetite for an ambitious shake-up of his Cabinet and other key advisers is already facing headwinds from inside his own administration and some Republicans on Capitol Hill.

Trump's nominee to run the CIA, Gina Haspel, is causing heartburn among Senate Republicans because of her role in overseeing an interrogation and detention program. Meanwhile, at least one Republican and a handful of Democrats in the narrowly divided Senate are already planning to oppose Mike Pompeo, Trump's nominee to succeed Rex Tillerson, who the President fired Tuesday from his post as secretary of state.

There is also some trepidation among the national security establishment about the potential of hardliner John Bolton replacing H.R. McMaster, whose fate as national security adviser is in doubt, according to several sources. And some top Senate Republicans are warning the White House of overburdening lawmakers with too many nominations.

"With everything else we have to do around here, having the prospect of two additional confirmation fights perhaps is going to be a challenge," said Sen. John Cornyn of Texas, the No. 2 Senate Republican.

In spite of that caution, Trump is signaling he's eager for a shake-up. He said Tuesday he is "getting very close to having the Cabinet" he has envisioned and has privately vowed to purge the "deadweight" amid a series of negative headlines that have engulfed about a quarter of his Cabinet, according to an administration official.

The plans for a broader shake-up are feeding the tumult that has engulfed the West Wing in recent weeks and has left officials on edge. Multiple White House sources told CNN they feel left in the dark and uneasy about who will be fired or will resign next. That uncertainty has contributed to low morale, which has plagued the White House for months.

Bracing for more

The West Wing is still bracing for more: McMaster could be on his way out and there are questions about Veterans Affairs Secretary David Shulkin and Housing and Urban Development Secretary Ben Carson. White House chief of staff John Kelly is also rumored to be departing at some point, though the exact timing is uncertain. The questions over Kelly's future have even led some of the aides closest to him to probe possible jobs outside the administration, one Republican close to the White House said.

Speaking to reporters Thursday in the Oval Office, Trump dismissed reports of a widespread shake-up, but said there "will always be change."

"I think you'll want to see change," he said. "I want to also see different ideas."

If the President is in a serious firing mood, the person most at risk could be Attorney General Jeff Sessions, who has endured ridicule, taunting and criticism far longer than any other Cabinet member. Yet his firing is more complicated than that of any other official.

For months, the President's friends and advisers have successfully urged him not to fire Sessions. But the President has not dropped the idea. A conversation is underway at the White House about whether EPA Administrator Scott Pruitt could become the next attorney general if Trump were to fire Sessions, according to a Republican familiar with the discussion.

While Pruitt's confirmation process could be brutal -- and uncertain -- he could be named acting attorney general. That position has to be filled by a Cabinet-level person who has already been confirmed by the Senate.

Pruitt, a former Oklahoma attorney general, was in the running last year to lead the Justice Department. Trump chose Sessions instead, a decision the President regrets, a senior White House official said.

The Pruitt scenario could unfold if Trump fired Sessions, which he has so far been unwilling to do.

Revolt among both parties

Several top Senate Republicans told CNN on Wednesday that firing Sessions would prompt a revolt among both parties and would reopen damaging questions about whether he was taking such steps to meddle with the Russia investigation. Republicans, who still have loyalty to their former Senate colleague, also have little appetite to deal with the controversy of a Sessions firing and rounding up the votes for a replacement.

"I would be surprised, disappointed and concerned" if Sessions were fired, Sen. Lindsey Graham of South Carolina, a Republican member of the Senate Judiciary Committee, told CNN. "It would be very difficult to find a replacement. It's a tough time right now to confirm somebody in a job like that. I think Jeff is doing a good job."

The committee chairman, Iowa Republican Chuck Grassley, warned Trump not to take the step of firing Sessions, but stopped short of saying his agenda was full and that he wouldn't hold confirmation hearings on a replacement.

Grassley told CNN last July that his agenda was "full" when rumors of a Sessions firing were percolating, a sign he had little desire to consider a replacement. Around that same time , Senate Majority Leader Mitch McConnell, a Kentucky Republican, also urged Trump not to fire Sessions, warning of an extraordinarily difficult confirmation process, according to a person briefed on their talks.

"I spoke on that a year ago and I don't have anything new to say," Grassley told CNN Wednesday. "And I don't want to say what I said six months ago holds today. But I would think it wouldn't be wise to fire Sessions."

Other vulnerable Cabinet members

Congressional Republicans have sought to protect other vulnerable members of the President's Cabinet from his ire, including Shulkin, who is facing reports that Trump is planning his ouster.

Sen. Johnny Isakson of Georgia and Rep. Phil Roe of Tennessee, the top Republicans on the Senate and House veterans committees, stepped out to praise Shulkin and express their confidence in him.

"I think Dr. Shulkin has done a bang-up job," Roe said. "I would certainly hate to see him leave that position."