Pete Hegseth, Trump's embattled Pentagon pick, appears to be gaining GOP support

President-elect Donald Trump's Cabinet picks were back on Capitol Hill on Tuesday looking to shore up support among Republican senators, including Pete Hegseth, his embattled choice for defense secretary.

Hegseth last week faced significant headwinds after new allegations of financial mismanagement, sexual impropriety and other disqualifying behavior were reported by the New Yorker. ABC News has not independently confirmed the magazine's report, and Hegseth's denied many of the accusations.

But he might be gaining back some ground. On Monday, a key Republican senator appeared to soften her view toward Hegseth.

Sen. Joni Ernst, a member of the Armed Services Committee and, like Hegseth, a combat veteran, previously said she wasn't ready to back Hegseth. But after their second meeting on Monday, she said she'd be "supporting him through this process" -- though she would not say whether she would ultimately vote in favor of his confirmation.

Hegseth said Tuesday he wouldn't speak for Ernst, but "I appreciate the time…. amazing conversation, foreign affairs, combat vet, welcomed all of her insight," he said while moving between meetings in Hart Senate Office Building.

Political pressure has ramped up on Republican lawmakers to back Trump's Pentagon choice, as a conservative group with ties to Trump ally Elon Musk is spending half a million dollars on a new ad pushing for Hegseth's confirmation. An ad urging viewers to call their senator and push them to back Hegseth will run on national television, but will also be targeted online in Iowa -- Ernst's home state.

ABC reported last week Trump was not making calls to senators on Hegseth, as he had for Matt Gaetz, his first choice for attorney general. But since then, the president-elect publicly backed Hegseth twice, once in a social media post and then in an interview with NBC that aired on Sunday. Trump told "Meet the Press" he still had confidence in Hegseth and that "he's going to do fine."

Then on Tuesday, the Trump team released a hype video for Hegseth that compiled friendly GOP senators praising him.

"Pete Hegseth is an exceptionally qualified, highly educated, and tough patriot who has not only served this country with courage, but has also devoted his life to advocating for our troops and veterans," Trump-Vance transition spokeswoman Karoline Leavitt said in a statement. "Mr. Hegseth's extensive military experience, combined with his steadfast commitment to restoring the Pentagon and our armed forces to greatness, will advance President Trump's vision of returning meritocracy, accountability, and efficiency to the United States military."

Multiple Republican senators said on Tuesday they were seeing a positive turn in the tide for Hegseth.

"Right now, he is in pretty good shape. I think he is very smart," Sen. Lindsey Graham said after last week calling the reports about Hegseth "very disturbing."

"So the accusations about mismanaging money and about non-consensual behavior. If they come forward, I will listen to those accusations, but they have to be credible and they have to be presented in a fashion that Pete can rebut. So, he's much better off this week than he was last week," Graham said.

Sen. John Cornyn, a key member of the Senate Judiciary Committee, told reporters after their meeting that he would support Hegseth during his confirmation.

"I will support Pete Hegseth, as I always say, barring any unforeseen circumstances. But I've known him a long time. I was satisfied with his answers and I think this is going to be a hard process on him and his family and some of it will be enormously unfair but I respect his willingness to give up a nice cushy job at Fox in order to serve the nation and the military that he has devoted a large percentage of his life to," Cornyn said.

Hegseth had a high-stakes meeting Tuesday afternoon with Alaska Sen. Lisa Murkowski, another Republican viewed as possibly opposing his confirmation.

Asked by ABC News Chief White House Correspondent Mary Bruce if she had learned anything that swayed her at all, Murkowski replied, "I had a good exchange with Mr. Hegseth."

Then, when asked by reporters if she would support Hegseth's nomination, she answered, "I had a good exchange and we'll see what the process bears."

She continued, "He's got probably at least half the Senate that he is going to visit with. He's got a process that is going to involve full vetting for the administration."

Hegseth told reporters, "I appreciate her very much. It was a wonderful meeting and a great discussion."

ABC News previously reported Trump's team was focused on determining where female GOP senators stand, given they could decide his fate given the party's thin three-seat majority in the new Congress. Hegseth is also expected to meet with Sen. Susan Collins on Wednesday.

Hegseth's agreed to an FBI background check, which many Republican senators said they wanted to be a part of the vetting process. He's also appeared to try to clean up his past comments about women in the military.

Hegseth previously said women should not serve in combat roles. During an interview on Fox News on Monday, Hegseth claimed he wanted to "clarify" his position, though did not explicitly reverse his stance.

"I also want an opportunity here to clarify comments that have been misconstrued that I somehow don't support women in the military," he said. "Some of our greatest warriors, our best warriors out there are women who serve, raise their right hand to defend this country and love our nation, want to defend that flag and do that every single day."

Other Trump picks on the Hill on Tuesday included Kash Patel, the president-elect's selection for FBI director, and Tulsi Gabbard, his pick for director of national intelligence.

Patel met with Sen. Markwayne Mullin, who afterwards voiced his support and told reporters he felt comfortable voting for him.

"If you're going to clean up the FBI and actually get them focused on mission rather than politics, refocus them on actually putting America first, doing their job, Kash is the perfect person for that," Mullin said.

ABC News' Benjamin Siegel and Kelsey Walsh contributed to this report.