After White House press secretary Karoline Leavitt said at Thursday's briefing it was President Donald Trump who fired CDC Director Susan Monarez, Monarez's lawyers pushed back that she has still not heard directly from the president on the matter.

Monarez's lawyers also said they have not had any further communications with the White House since Wednesday night, when a White House staffer notified her of her termination, which her attorneys then rejected as "legally deficient."

"WH @PressSec can say whatever she wants because thankfully free speech still exists in this country. But it doesn't make her comments factually true, even when from a White House podium," attorney Mark Zaid said on X after the press briefing.

Zaid and attorney Abbe Lowell have said they would only accept termination from the president given Monarez's status as a Senate-confirmed appointee.

"Our client was notified tonight by a White House staffer in the personnel office that she was fired. As a presidential appointee, Senate-confirmed officer, only the president himself can fire her. For this reason, we reject the notification Dr. Monarez has received as legally deficient and she remains as CDC Director. We have notified the White House Counsel of our position," the attorneys said on Wednesday night.

-ABC News' Cheyenne Haslett

White House press secretary Karoline Leavitt was asked if President Donald Trump believes the COVID vaccine should be available to and covered by insurance for all Americans regardless of age or preexisting conditions.

Leavitt said the FDA revoked emergency youth authorizations for some COVID vaccines, but that the decision "does not affect the availability of COVID vaccines for Americans who want them."

"We believe in individual choice. That's a promise both the president and the secretary have made, and it's a promise they have now delivered," she said.

The White House press secretary was asked why President Donald Trump ordered Fed Governor Lisa Cook fired when she had not been charged with a crime and the allegations of mortgage fraud were still being probed.

Leavitt noted that Cook is challenging the firing and admitted that the Justice Department is still probing the allegations, but backed the president's actions without giving more details.

"You had these mortgage receipts, very clearly shown to the president, and he has the cause that he needs to fire this individual. He laid out in the letter that he provided to her and to the public as well. And so we'll continue to fight this battle," she said.

White House press secretary Karoline Leavitt, at Thursday's press briefing, discussed the ongoing investigation into the shooting at Annunciation Catholic School in Minneapolis, Minnestoa.

One reporter asked if the administration believed the shooter's gender identity should be part of that investigation.

"All of the elements are being investigated in the shooting. Absolutely. Which is why part of the reason the FBI director came out and immediately confirmed that facet of the investigation," Leavitt said.