Jared Kushner, President Trump's son-in-law and senior adviser, is under consideration to become the next White House chief of staff, CBS News chief Washington correspondent Major Garrett has confirmed, citing a source familiar with the process.

Kushner met with Mr. Trump on Wednesday to discuss the job. Kushner's candidacy for chief of staff was first reported by HuffPost.

Current White House chief of staff John Kelly is set to leave at the end of the year. Mr. Trump told reporters Thursday there are five "well-known" and "terrific" people on his short list of candidates to replace Kelly.

"We're interviewing people," he said.

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North Carolina Rep. Mark Meadows withdrew himself from consideration for the chief of staff post on Wednesday. "Congressman Mark Meadows is a great friend to President Trump and is doing an incredible job in Congress," White House press secretary Sarah Sanders said. "The president told him we need him in Congress so he can continue the great work he is doing there."

Meadows had been a contender for the role, after Vice President Mike Pence's chief of staff, Nick Ayers, dropped out of the running. Ayers and Mr. Trump couldn't come to an agreement on a time commitment for the post, and the president was forced to revisit other options.

White House Deputy Press Secretary Hogan Gidley said Friday that Kelly may agree to extend his time as chief of staff.

"Obviously, if the president and the chief of staff make another deal and extend it, they can do that. It's their prerogative to do so. Right now, currently, John Kelly is expected to leave at the first of the year," Gidley said.