Mr. Parnell withdrew from a Senate race in November 2021 after a judge ruled that his estranged wife should have primary custody of their three children in a case in which she accused him of spousal and child abuse. The order was later vacated and Mr. Parnell has shared custody with his wife for several years, a Pentagon official said.

Mr. Parnell, who spoke at the 2024 rally in Butler, Pa., where Mr. Trump was nearly assassinated, has a strong relationship with the president, officials said. But senior administration officials said Mr. Parnell likely would struggle to be confirmed by the closely divided Senate.

Mr. Driscoll, a 40-year-old Army former first lieutenant, was until recently seen as a potential candidate to replace Mr. Hegseth if he decided to step down or fell out of favor with Mr. Trump. He garnered more support on Capitol Hill than did Mr. Hegseth, who has a combative relationship with the Democrats and even some Republicans on the key House and Senate committees that oversee the Pentagon.

Mr. Driscoll, in contrast, was viewed by many in Congress as one of the few Trump administration Pentagon officials who can work with lawmakers in both parties, two congressional aides said. While Mr. Hegseth has tried to limit direct communications between military officials and lawmakers, Mr. Driscoll has responded to requests from Republicans and Democrats alike in the Armed Services Committees, the aides said.

During his nomination hearings, Senator Elizabeth Warren, Democrat of Massachusetts, pressed him to commit to allowing the Army to repair its own equipment, instead of sending it back to defense contractors. Months later, Mr. Driscoll did so, putting the policy, known as “right to repair,” in place and earning a rare plaudit for a Trump administration official from a Democratic lawmaker (and one who had voted against his nomination).

But Mr. Driscoll told Mr. Trump that he was frustrated with Mr. Hegseth’s ouster of so many proven Army leaders and his blocking the promotions of a number of colonels who had worked for people he did not like, two officials said.