"Pete is doing very well," the US president-elect said on his Truth Social platform, in his first public comment on the matter since Hegseth began a round of visits seeking Senate support.

In his post, Trump called Hegseth "a WINNER," adding that "there is nothing that can be done to change that!!!"

Amid reports that the Hegseth nomination is in peril, Trump seemed intent on shoring up Republican support for the 44-year-old military veteran as his pick to head a department with three million employees and a defense budget surpassing $800 billion.

Top presidential nominations require confirmation by a majority of the Senate. And according to NBC News, perhaps six Republican senators are lining up against Hegseth -- enough to scupper his nomination in a chamber where Republicans hold a slim 53-47 majority.

Since his nomination was announced, Hegseth has been busy visiting senators, especially those thought to be wavering, to lobby for support.

"We're going to earn those votes," he said Thursday in the Capitol.

And on Friday he thanked Trump for his continued support, saying on X that "Like you, we will never back down."

Just days after Hegseth's nomination was announced, accusations emerged of a sexual aggression dating from a 2017 encounter in California.

But no complaint was filed at the time, and the former National Guards officer -- he served in both Iraq and Afghanistan -- has denied any non-consenting relationship.

But doubts grew after the New York Times reported that Hegseth's own mother in 2018 had said in an email to him that he was an "abuser of women" and lacked character.

Penelope Hegseth later appeared on Fox News, the conservative news channel where her son worked until last month as a commentator, to say the email had been written "in haste" during his divorce.

The New Yorker then reported that Hegseth had resigned as head of two veterans' groups following accusations of sexual aggression, excessive drinking and financial mismanagement.

Trump has already seen one nominee forced out: former lawmaker Matt Gaetz withdrew as the president-elect's choice for attorney general following accusations of sexual relations with a minor. Gaetz has denied any wrongdoing.