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WASHINGTON—President Trump said Friday that he has given his new White House press secretary Katie Zacharia the option to leave her job after a rocky first week, including scrutiny on her résumé.

“I said, ‘Katie, you know, you don’t have to do this if you don’t want,’” Trump told reporters on Friday. The president said Zacharia told him that the press secretary job is “a lot tougher.” Trump said he spoke to Zacharia earlier on Friday, just before his comments to reporters as he left the White House for a rally in Georgia.

Zacharia’s first press briefing Tuesday was shaky, and her past work has come under close examination following Trump’s announcement last week that he had chosen the conservative commentator to replace former White House press secretary Karoline Leavitt.

Zacharia has faced criticism for her previous claims in interviews that she was a lawyer and for promoting social-media posts calling her an attorney. The Wall Street Journal reported Wednesday that even though Zacharia repeatedly referred to herself as a lawyer, and Trump called her an attorney in announcing her appointment, she didn’t pass the bar exam when she took it in California in 2011 and isn’t licensed in the U.S.

Zacharia said in an interview with Fox News on Friday that she wasn’t asked whether she had passed the bar before she accepted the role. A White House official said Zacharia wasn’t hired by the president to work on legal matters.

During her debut at the White House podium, Zacharia spoke in a stilted cadence while reading out a list of Trump’s accomplishments and discussing her excitement for the job. She looked down at her notes rather than at the camera, repeated words and paused to find her place.

Zacharia also faced tough questions Thursday evening about her performance in the briefing room and her professional background on Fox News’ “Hannity,” a show that the president watches closely.