As lawmakers on Capitol Hill are racing to avoid a government shutdown next week, President-elect Donald Trump's cabinet nominees continue to make their case to Senators who could service hurdles to confirmation.Sen. John Fetterman of Pennsylvania on Thursday became the first Democrat to meet with Pete Hegseth, Trump's pick for secretary of defense. But it remains unclear how that meeting went, as Fetterman declined to share whether he would support Hegseth for the position.RELATED STORY | Trump doubles down on support for Hegseth amid contentious nomination process"We had a conversation and that's part of the process," Fetterman told Scripps News. "And that's the thing, I'm going to follow the process and I'm going to listen to what my colleagues on the other side continue to say and how they evaluate."For a while it appeared that Hegseth's nomination was unstable. But in the past couple of days it seems like he may have rebounded.Hegseth has also met with a number of other senators who could be pivotal in his confirmation, including Sen. Susan Collins of Maine a key moderate Republican in the chamber and GOP Sen. Lisa Murkowski of Alaska. Both women came out of meetings with Hegseth saying they were good conversations, but refused to say whether they would support him.RELATED STORY | Mother of Pete Hegseth reportedly told son he was an 'abuser of women' in emailHegseth's past was thrust into the spotlight following news of his nomination to lead the Department of Defense. He has been accused of sexual misconduct and mismanagement of funds while leading a veterans' charitable organization. Questions also still remain about alcohol and drinking issues that Hegseth reportedly had while working at Fox News.Republicans have a large enough majority in the next Congress to not need Democratic votes to confirm any of Trump's cabinet nominations but that majority is still slim enough that the party can sustain only a few defections.