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(CNN) —  

A day after his labor secretary convened a news conference defending himself against accusations of prosecutorial malfeasance, President Donald Trump is questioning whether Alex Acosta’s efforts were sufficient in quieting the uproar, people familiar with the matter said.

Trump has remained publicly silent in the 24 hours since Acosta held the question-and-answer session – at the President’s urging – to defend his role in financier Jeffrey Epstein’s plea agreement over sexual assault allegations, which critics have bemoaned as woefully lenient.

Epstein was ultimately sentenced to time behind bars – but served only 13 months and was allowed to leave for long stretches to work from his office. He avoided a federal trial but did have to register as a sex offender.

Trump, who a day before described Acosta as a “really great” Cabinet member, has privately expressed concern at how the labor secretary’s handling of the Epstein case reflects on him and his administration. Trump has not appeared overly concerned with the plea agreement itself, people close to him said.

The President and his aides have been looking at some of the legal commentary since Acosta’s performance and are waiting to see how things develop, one White House official said, adding that some in the West Wing want to see whether a review of Acosta’s actions reveal any other concerns.

“He’s not out of the woods yet by any stretch,” the official said.

In his news conference, Acosta described securing the best deal he could when he was a US attorney in Miami. He claimed that taking the case to trial would be have been a risk, and said his main priority was ensuring Epstein served time in jail and registered as a sex offender.

Aides worked Wednesday and Thursday to convince Trump that Acosta performed well during his news conference, which Trump did not watch from start to finish. The officials told Trump that Acosta suitably answered questions about his role in Epstein’s plea agreement.

Trump initially appeared receptive and told people he believed Acosta performed well, according to the people familiar with the matter.

One aide said Acosta’s methodical description of events would provide Trump’s allies ample ways to push back on criticism from Democrats and others who have been harsh in their assessment of Acosta’s handling of the case.

Since then, however, Trump has begun quizzing people around him whether Acosta had done enough to quiet the uproar, appearing skeptical the question-and-answer session would end the controversy, people familiar with the situation said.

Because Trump has yet to publicly state his views, some officials have continued to speculate about Acosta’s future.

In his news conference, Acosta described his relationship with Trump as “outstanding.” But he acknowledged that could change.

“I’m doing my job. If at some point the President decides that I am not the best person to do this job, I respect that. That is his choice,” he said. “I serve at the pleasure of the President.”

CNN’s Jim Acosta contributed to this report.