President-elect Donald J. Trump’s support for Pete Hegseth, who he announced as his nominee for defense secretary shortly after Election Day, is wobbling after a crush of controversy over a rape allegation and a 2018 email from Mr. Hegseth’s mother accusing him of a pattern of abuse toward women.

How Mr. Hegseth fares through a series of tests on Wednesday will be critical for his chances. He is set to meet with key senators, including Joni Ernst of Iowa, a combat veteran who has spoken about being sexually assaulted herself, and his mother is expected to sit for an interview on Fox News. He is also set to start defending himself on television.

Mr. Trump has made clear to people close to him that he believes Mr. Hegseth should have been more forthcoming about the problems he would face getting confirmed, according to two people with knowledge of his thinking.

The combination of events could determine whether he hangs on as the expected nominee. Mr. Trump is openly discussing other people for the job, including Gov. Ron DeSantis of Florida, whom he defeated in the Republican presidential primaries and with whom he has had a contentious relationship. Mr. Trump likes the story of bringing on someone he dominated publicly, and he talked about it with Mr. DeSantis yesterday at a service honoring three Florida sheriff’s deputies who were killed in a car crash.

But the number of people in Mr. Trump’s world who dislike and distrust Mr. DeSantis — and bitterly recall the campaign he ran against the president-elect — is vast. Those people are discussing other options, including whether Mike Waltz, the Florida congressman who Mr. Trump picked as his national security adviser, could slide into the job, expecting he would be confirmed fairly easily by the Senate.

“I think some of these articles are very disturbing,” Senator Lindsey Graham of South Carolina, a close Trump ally, told CBS News, referring to Mr. Hegseth. “He obviously has a chance to defend himself here, but some of this stuff is going to be difficult.”

Mr. Trump has spent little if any personal capital with senators trying to push Mr. Hegseth through. And the incoming president’s advisers are mindful in private discussions that Republican senators are trying to be respectful of Mr. Trump while not approving of a nominee who concerns them.

Mr. Hegseth, 44, could become the third person whom Mr. Trump has announced as a nominee to withdraw from the role after Matt Gaetz withdrew his name for attorney general and Sheriff Chad Chronister withdrew as D.E.A. administrator.

In the past two weeks, Mr. Hegseth has come under intense scrutiny. It was revealed that he had entered into a settlement agreement with a woman who accused him of rape in 2017; he had insisted it was a consensual encounter, and Mr. Trump told aides at the time that he wanted to stick with his announced nominee.

But the troublesome headlines, which Mr. Trump hates, only grew worse. The New York Times reported on an email his mother wrote him in 2018 as he was going through an acrimonious divorce, in which she told her son he had “abused” a number of women “in some way” over the years. Mr. Hegseth’s mother has recently said she regretted sending the email and has retracted the comments that she said she made during an emotional time as he was going through a divorce — a comment she is expected to expand upon in her Fox appearance.

Mr. Trump has told people he was unhappy with the story about the email. It was unclear how extensive the vetting was into Mr. Hegseth’s past by Mr. Trump’s transition team.

Now the Trump team will watch closely how Mr. Hegseth and his mother perform in the interviews, knowing they will be critical for the incoming president in deciding whether to stick with the former Fox News host and combat veteran whose qualifications to lead the Pentagon have come into question.

The perception from people close to Mr. Hegseth is that if he wants to save himself, he must perform well. The Trump team is particularly worried about female Republican senators breaking with Mr. Hegseth, and especially Ms. Ernst.