Mr. Ayers tweeted that he would instead be leaving the White House soon, apparently to return to his home state of Georgia.
A person familiar with Trump's thinking tells The Associated Press that the director of the Office of Management and Budget, Mick Mulvaney, is among those under consideration to replace John Kelly.
Nick Ayers, who is now the U-S vice president's chief of staff, has been mentioned as a potential replacement.
CNN has also confirmed Meadows is on the shortlist to replace Kelly, who is expected to leave by the end of the year.More news: Brexit: Theresa May in last-ditch appeal to Brussels to avoid defeat
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On Sunday, citing unnamed White House officials, the Wall Street Journal said Trump and Ayers had failed to agree on terms for the job. Ayers also earned the backing of Ivanka Trump and Jared Kushner, the president's daughter and son-in-law and senior advisers, for taking on the new role, White House officials said.
That leaves it unclear who will take the top West Wing job as Trump enters the 2020 campaign confronting growing danger from the special counsel investigation into Russian election interference and from Democrats who have vowed tougher oversight after they take control of the House next month. Ayers backing out will slightly delay Mr. Trump's decision, which will now be announced by the end of the year instead of this week. Mr. Trump, who has denied collusion, calls the probe a "witch hunt".