Office of Management and Budget Director Mick Mulvaney, North Carolina Rep. Mark Meadows and longtime GOP operative Wayne Berman are among the contenders for the job of White House chief of staff, as President Trump considers his list of potential replacements for John Kelly. He has a list of five candidates at the moment, according to a senior white House official.

In a tweet Tuesday morning, Mr. Trump said "many, over ten, are vying for and wanting the White House Chief of Staff position. Why wouldn't someone want one of the truly great and meaningful jobs in Washington?"

"We're in no rush," the president said in response to a question Tuesday afternoon from reporters in the Oval Office. "I have some great people — a lot of friends of mine want it," he said, adding that the job is being handled now by Kelly, "a wonderful chief of staff." A decision, Mr. Trump said, will be made in a "week or two, or maybe less."

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Walking into a Republican conference meeting on Tuesday morning on Capitol Hill, Meadows said that he has had no new conversations with the White House regarding the job. He told reporters Monday that any opportunity to serve the president would be an honor.

Meadows has been serving as the House Freedom Caucus chairman and already has a strong relationship with the president.

"Any time that your name is floated to help the president of the United States accomplish things for the American people, it's flattering," Meadows said outside his office Monday. "I'm not out campaigning for the job."

Later, in an appearance on Fox News' "The Story," Meadows told host Martha MacCallum, "My life changed 24 hours ago when (Nick) Ayers decided he didn't want it." Ayers, currently Vice President Pence's chief of staff, turned down the job after he and Mr. Trump could not reach an agreement on the terms. Meadows also said that he's "favorably inclined to have a conversation with the president," but he has not spoken with Mr. Trump since Ayers removed himself from the running.

The need for a new chief of staff arose after Mr. Trump abruptly announced Saturday Kelly will be leaving the administration "at the end of the year," a move he made without mentioning it once during a dinner with senior staff the night before, according to a senior administration official.

One source close to the conversations around a new chief of staff tells CBS News that President Trump is "pissed off" at Ayers for not signing onto the long-term job, an unexpected blow that effectively forced Mr. Trump to restart his search. After Ayers leaves the administration at the end of the year, he will take a job at a pro-Trump super PAC.

Sources said Ayers had offered to serve as interim chief of staff, but Mr. Trump wanted him to commit to serving for least two years, which Ayers declined. The 36-year-old father of three young triplets will instead move back to his native Georgia, as he had long planned before the chief of staff job opened up.

"Thank you @realDonaldTrump, @VP, and my great colleagues for the honor to serve our Nation at The White House," Ayers tweeted. "I will be departing at the end of the year but will work with the #MAGA team to advance the cause. #Georgia."

One name that surfaced on Monday was Wayne Berman, according to a person familiar with the ongoing process. Berman, head of global government affairs for the investment firm Blackstone, served as chairman of Sen. Marco Rubio's 2016 presidential campaign, as a senior adviser to the Romney-Ryan 2012 Republican presidential campaign, and was also national finance chairman for Sen. John McCain's 2008 presidential bid. He once served as an assistant secretary of commerce during George H.W. Bush's administration.

In the world of Trump personnel moves, Berman perhaps best fits into the "often mentioned, never picked" category. He has no direct personal or professional connection to Trump, which some see as critical for taking on such an important role. But during the 2016 presidential transition, he was mentioned for potential Cabinet or senior government positions, and Berman was touted as a potential successor to Reince Priebus when reports about Priebus' eventual departure ass chief of staff began in 2017.

Meanwhile, Mulvaney, the OMB director, has made clear to the president that he wants the job on a full-time basis, not as a placeholder. "Talks broke down last night when Trump talked with him about interim [chief of staff] not full time and he doesn't want the interim title," said someone who spoke with Mulvaney on Monday morning. CBS News chief White House correspondent Major Garrett reported last month that Mulvaney and Treasury Secretary Steve Mnuchin were being considered for the post.

Another person close to several administration officials who is keeping close tabs on the process from outside the White House described the ongoing search as "chaos." "The problem is Trump is chief of staff and over his head," said the person, who was granted anonymity to speak frankly about ongoing deliberations. Ultimately, the White House is looking for "a caretaker or a person that can manage the Cabinet, etc."

In a late tweet Sunday night, Mr. Trump said, "I am in the process of interviewing some really great people" for the chief of staff role and added that he would be "making a decision soon!" His announcement is expected to come by the end of the year.

Arden Farhi, Richard Escobedo and Sarah Horbacewicz contributed to this report.