Donald Trump has for weeks been privately testing the idea of replacing his chief of staff, Mick Mulvaney, who's swiftly fallen out of favor with some of the president's allies after high-profile stumbles handling the House impeachment inquiry.

About a month ago, Trump said to Treasury Secretary Steven Mnuchin in front of a roomful of staff: You have such great ideas, why don't you be my chief? He has made similar remarks about Chris Liddell, a deputy chief of staff at the White House, according to people familiar with the matter. He's also asked advisers whether his counselor Kellyanne Conway would be a good chief of staff, other people said.

Some White House aides say this is nothing new, and that Trump often tests similar ideas in conversation or makes such remarks to flatter his aides and keep others on their toes.

Mnuchin is not under formal consideration for chief of staff, one person familiar with the matter said. He is one of Trump's most loyal aides, but his removal from the Treasury Department could disturb markets, where investors have come to regard him as a source of stability in an otherwise volatile political environment.

But Trump's musings about replacing Mulvaney are a sign of the president's growing discontent with his performance. For almost a year, Mulvaney has served as "acting" chief of staff because Trump has withheld the permanent title from him.

Mulvaney's standing with his boss has become unsteady after the White House's flat-footed response to the impeachment inquiry, which some Trump allies have blamed on the chief of staff. Mulvaney further damaged himself with a news conference last week in which he admitted that aid to Ukraine was held up to pressure the country to investigate the president's political rivals — the question at the heart of the impeachment debate.

Previous staff shake-ups have begun with Trump asking aides and associates whether this person or that should be replaced. Former Attorney General Jeff Sessions, for instance, endured extended public speculation about his departure, fueled by Trump himself. Mulvaney is the president's third chief of staff, appointed by tweet in December to replace John Kelly, who himself was appointed by tweet to replace Reince Priebus.

Trump didn't respond on Monday when he was asked during a Cabinet meeting whether Mulvaney would remain in his job. One aide said that the president's silence was a signal that Trump is considering a change.

Mulvaney's departure isn't seen as imminent. But some of Trump's closest associates have assembled a roster of possible replacements if the president decides to replace Mulvaney, Bloomberg News reported Sunday. Among those said to be on the list are former acting Attorney General Matthew Whitaker, former New Jersey governor Chris Christie, and veteran political operative Wayne Berman, now a senior managing director for government relations at the Blackstone Group.

Some senior White House staff have rallied around Mulvaney. On Monday, Mulvaney admitted during a senior staff meeting that he had erred somewhat in the news conference, according to two people familiar with the matter. Other officials responded with support and a round of applause.

"Mick Mulvaney's standing in the White House has not changed," White House spokesman Judd Deere said Monday in an e-mail. "He is still the acting chief of staff and has the president's confidence."

Mulvaney has a number of influential allies in and close to the White House, two people said, including Russ Vought, acting director of the Office of Management and Budget; Joe Grogan of the White House's Domestic Policy Council; and Patrick Pizzella, Deputy Secretary of Labor.

And as Trump tries to retain his political support among Senate Republicans, who are expected to have to vote on his impeachment, some allies have raised concerns about his chief of staff.

Sen. Chuck Grassley told Politico that Mulvaney appeared at the news conference to be "somebody that didn't know what they were talking about."

Christie was a top contender for the White House's top job after Kelly was ousted late last year. But the former New Jersey governor withdrew from consideration in part over concerns that a book he planned to publish would create headaches for the White House. The memoir said that Jared Kushner, the president's son-in-law and one of his top advisers, forced Christie from his job leading Trump's transition team over a long-held resentment for Christie's prosecution of Kushner's father, Charles, in 2005.

Kushner would also likely object to Conway's appointment as chief of staff, two people familiar with the matter said — the two White House aides aren't friendly.

Some Mulvaney allies have meanwhile recently criticized White House Counsel Pat Cipollone, another potential replacement. They say the attorney is to blame for the administration's impeachment strategy, and described Cipollone as having a bunker mentality that's kept the White House on lockdown and discouraged allies from publicly fighting for the president.

Mulvaney said Sunday he hasn't offered his resignation to Trump over Thursday's press briefing.

"Did I have the perfect press conference, no," Mulvaney said on "Fox News Sunday." "I still think I'm doing a pretty good job as the chief of staff, and I think the president agrees."

One White House official concurred, saying that Trump was pleased with Mulvaney's performance at the briefing and asked that the message be passed along to the chief of staff. And the president's campaign seemed to offer Mulvaney a signal of support late last week, announcing the sale of T-shirts featuring a phrase Mulvaney used in the news conference to dismiss concern about Trump's intermingling his political interests with U.S. foreign policy: "Get over it."

But Mulvaney has come under fire from some of the president's loudest cheerleaders, including Fox News personality Sean Hannity, who called the acting chief of staff "dumb" during his Friday radio show and said his "interpretation of things" was "idiotic." Fox Business Network host Lou Dobbs called Mulvaney's press conference "thoroughly confusing" and said it "seemed to contradict what the president has said."

Both broadcasters frequently laud the president, who often returns the praise. But Hannity didn't ask about the chief of staff in an interview with Trump he aired on Monday.