WASHINGTON, D.C. – President-elect Donald Trump's pick for defense secretary, Minnesota native Pete Hegseth, said he has the former president's continued support amid reports his nomination may be in jeopardy.

"I spoke to the president this morning, he supports me fully. We're not going anywhere," he told reporters Wednesday on Capitol Hill after meeting with Republican senators throughout the day.

His remarks came after multiple reports Trump may be having second thoughts about naming the Forest Lake native as defense secretary and could turn to Florida Gov. Ron DeSantis to fill the role.

An increasing number of lawmakers have raised concerns Hegseth may face a difficult time getting through the confirmation process after allegations of sexual assault and alcohol abuse have come to light.

Sen. Kevin Cramer, R-N.D., who sits on the Senate Armed Services Committee, which will weigh Hegseth's nomination, said he would like to see Hegseth pledge not to drink. "It would help me a lot," he told reporters Wednesday.

Asked whether he thinks Hegseth will make it through the confirmation process, Cramer said he wasn't sure. "I think people are ready to give him the benefit of the doubt if they can have commitments from him about things going forward," he said.

A simple majority of committee members would need to approve his nomination before the full Senate would vote. If Hegseth makes it out of the committee, it would take just four Republican senators voting against him to sink his chances.

Several senators on and off the Armed Services Committee were not ready to fully commit to supporting him when asked Wednesday.

"I'm going to meet with him next week. I've got lots of questions," said Sen. Lisa Murkowski, R-Alaska. Murkowski is a moderate Republican who voted to convict Trump for his role in the Jan. 6, 2021, U.S. Capitol attacks and could join Democrats in voting against Hegseth.

Sen. Tom Cotton, R-Ark., who also sits on the Senate Armed Services Committee, said "no comment" when asked if he plans to support Hegseth.

"The president makes that decision, we don't," said Sen. Mike Rounds, R-S.D., another committee member, when asked whether he would prefer another defense nominee. "We give the benefit of the doubt to the president, then we try to do our best to help him get his team in place."

Some Republicans are standing firmly behind Hegseth.

"I still 100% support him," Sen. Markwayne Mullin, R-Okla., said Wednesday.

"The window is still open for him and he can make it happen," Mullin continued when asked if he thinks Hegseth will make it through the confirmation process. He called the possibility of other nominees "what ifs."

The Wall Street Journal was first to report that Trump may not appoint Hegseth, a former Fox News co-host, former Army National Guard major and a combat veteran who was deployed to both Iraq and Afghanistan, as his secretary of defense.

Former Minnesota GOP Sen. Norm Coleman has joined Hegseth on Capitol Hill this week to help him navigate the Senate and meet with senators ahead of a potentially contentious confirmation process.

Brian Melendez, who was Minnesota DFL party chair when Coleman served in the Senate, said the last Republican senator from Minnesota is an obvious pick to lead the Minnesota nominee through the process, although Coleman hasn't been in the Senate since 2009.

"He knows how the process works and he knows how to talk to people, so that adds some value right there," Melendez said.

Hegseth, a Forest Lake High School graduate, has denied the assault allegations and other reports about his work conduct and history. He said he's willing to meet with senators who are potentially skeptical of his nomination.

''We're going to meet with every senator that wants to meet with us, across the board,'' Hegseth told reporters on Tuesday. ''And we welcome their advice as we go through the advice and counsel process.''

This story contains material from the Associated Press.