With the public firing of Secretary of State Rex Tillerson this week, the Trump White House appears to be doing some spring cleaning, and may remove other administration figures from their posts in the coming days. "[T]he president wants to make sure that he has the right people in the right places for the right time," White House Press Secretary Sarah Huckabee Sanders said on Thursday.

As CBS News' chief White House correspondent Major Garrett reports, in the Trump administration, nothing is certain until it happens, but there are rumblings that President Trump isn't yet finished with his staff shakeup.

Here are the Cabinet members that are at risk of meeting Tillerson's fate:

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National Security Adviser H.R. McMaster

National Security Adviser H.R. McMaster could be the next top administration official to go, as CBS News chief White House correspondent Major Garrett has reported. If McMaster is ousted, he is expected to be replaced by former Bush administration official and frequent Fox News guest John Bolton, who is known for his hawkish views on Iran and North Korea.

Speculation over McMaster's future has ensued for weeks, and he's made public comments putting him at odds with the commander-in-chief. McMaster told the New York Times in December that Mr. Trump's approach to foreign policy "has moved a lot of us out of our comfort zone, me included."

Earlier this year while speaking overseas, after special counsel Robert Mueller indicted 13 Russians for influencing U.S. elections, McMaster said evidence of Russian meddling is "beyond dispute."

Mr. Trump didn't take that well, tweeting shortly thereafter that McMaster, "forgot to say that the results of the 2016 election were not impacted or changed by the Russians and that the only Collusion was between Russia and Crooked H, the DNC and the Dems. Remember the Dirty Dossier, Uranium, Speeches, Emails and the Podesta Company!"

Housing and Urban Development Secretary Ben Carson

Congressional sources and White House advisers tell CBS News that Housing and Urban Development Secretary Ben Carson is in serious jeopardy. There is an expectation he will not last the week in the Trump administration.

Carson's troubles largely stem from the $31,000 in furniture ordered by Carson's office last year, a fact detailed extensively in emails released by the watchdog group American Oversight on Wednesday. This hasn't made the president happy. A spokesman for the department initially said Carson had no involvement in the lavish purchase. But internal emails claimed Carson and his wife, Candy Carson, selected the pieces themselves.

The emails also revealed HUD employees devoted months of staff time to remedying Carson's furniture situation, and there was pressure to find a workaround to exceed the $5,000 statutory limit for refurbishing his office.

Asked about the purchase on Thursday, White House press secretary Sarah Sanders only said that the matter is being reviewed.

"This is something we're looking into — I don't have any updates at this time," Sanders said.

Veterans Affairs Secretary David Shulkin

Veterans Affairs Secretary David Shulkin could be fired by President Trump and replaced with Energy Secretary Rick Perry.

The threats of a reshuffling comes after months of turmoil for Shulkin, who was subject of an internal watchdog investigation surrounding the secretary's travel expenses for a trip to Europe last year.

The investigation found that Shulkin had improperly accepted Wimbledon tennis tickets and likely used taxpayer money to cover his wife's airfare for an 11-day European trip. The report also questioned Shulkin's decision to direct agency staff on official time to arrange personal sightseeing activities for him and his wife during the July trip to England and Denmark.

Shulkin said he realized the "optics are not good" and planned to reimburse the Treasury Department.

A more recent watchdog report faulted "failed leadership" and a "climate of complacency" for putting patients at risk at a D.C. VA hospital while Shulkin was serving as the department's under secretary during the Obama administration.

The report said that at least three program offices had sufficient information to inform Shulkin of prevalent safety issues at the medical center but Shulkin took no action to resolve the problems.

The secretary said that he "does not recall" senior leaders bringing the issues to his attention and had only found out of the systemic issues about a year ago.

Perry has since denied having in interest in Shulkin's job, saying he's staying put at Energy "until the foreseeable future -- happily."

Sanders said at Thursday's press briefing of the ongoing turmoil at the VA: "The president has a large number of individuals that are working to make sure that the VA is helping veterans at the best level possible. We are continuing to review if we can improve on that system."

Education Secretary Betsy DeVos

President Trump's Education Secretary Betsy DeVos came under widespread criticism for her lackluster response to questions from 60 Minutes' Lesley Stahl about schools in her home state of Michigan.

DeVos struggled with basic questions about how Michigan is dealing with underperforming public schools. Asked by Stahl if she had seen the schools herself to figure out how to best address the ongoing issues, DeVos replied "I have not, I have not intentionally visited schools that are underperforming."

White House press secretary Sarah Huckabee Sanders said she's not sure if the president saw the entire interview. But sources tell CBS News' Major Garrett that administration officials inside and outside the White House are regarding the interview as uneven and ill-prepared.

Her poor performance follows months of similar blunders — most notably calling historically black colleges and universities "pioneers of school choice." She is frequently met by protesters and hecklers for championing school choice proposals.

However, for now she still has a vote of confidence from the president, who put DeVos in charge of a commission that will focus on school safety in the wake of the deadly school shooting in Parkland, Florida.

Chief of Staff John Kelly

One of Mr. Trump's closest advisers, John Kelly, is potentially in the crosshairs of the president's staff shake-up. According to congressional and administration sources, Kelly, who was made chief of staff over the summer in an attempt to bring order to the West Wing, may be on the way out.

Kelly himself had reportedly offered to step down over the way he handled the allegations of domestic abuse against now-former staff secretary Rob Porter. He told reporters after Porter's departure that the White House "didn't cover ourselves in glory" during the situation, but later said there was "nothing to even consider resigning over"

Attorney General Jeff Sessions

President Trump has publicly derided his attorney general for months. The president has expressed frustration over Session's decision to recuse himself in the Russia probe — a decision Sessions has stood by — and over what sees as Sessions' failure to investigate the president's former opponent, Hillary Clinton.

But Sessions has also been in charge of implementing policies Mr. Trump campaigned on, such as cracking down on illegal immigration and the "sanctuary cities" that refuse to cooperate with federal immigration authorities.

CBS News' Kathryn Watson, Emily Tillett, Major Garrett and Jacqueline Alemany contributed to this report.