Admiral Howard would bring a lengthy and detailed résumé of service in the armed forces to an agency that was plagued by controversy during President Obama’s tenure. She would fit with Mr. Trump’s affection for military officers — he has named two generals to his cabinet so far and one as his national security adviser. She would also be the second African-American to join the cabinet.

Aides to Mr. Trump said that they were getting many suggestions and that the pick for the V.A. would take some time. Mainline veterans groups have asked the president-elect to keep the current secretary, Robert A. McDonald, a former Procter & Gamble chief executive. He is seen as turning the agency around after a series of scandals under his predecessor, Gen. Eric K. Shinseki.

But people in the Trump circle like the idea of a military figure with management experience.

Another Fox commentator for the N.S.C.?

Monica Crowley, the Fox News commentator who worked with President Richard M. Nixon on his last two books, is likely to be named as a strategic communications adviser to the National Security Council, according to two people close to the transition.

The appointment would put Ms. Crowley in the sphere of influence led by Lt. Gen. Michael Flynn, the president’s pick for national security adviser. Ms. Crowley had been mentioned as a possible pick for White House press secretary early in the process.

She would join K.T. McFarland, another Fox personality, who has been chosen to be General Flynn’s deputy.

Trump tweets. Decades-long vendetta continues.

He may be about to become the leader of the free world, but Mr. Trump still holds a grudge against Graydon Carter that started in the days of Spy magazine and that continues with the magazine Mr. Carter now edits, Vanity Fair.

After holding his fingers back for a few days, the president-elect went on Twitter again Thursday.