US media have reported there are calls to replace Hegseth - in particular over his handling of the Iran war, which is being blamed for heavy Republican election losses.
One outgoing congressional Republican has threatened to force an impeachment vote against Hegseth as early as next week. The defence secretary posted a video statement defending his record and denouncing the effort.
"You see, they attack me. They attack us because we are effective," he said. "It's not going to work. We will not back down."
Meanwhile, Rubio is denying reports he is considering leaving his post ahead of a 2028 presidential run.
Working for Trump until the end of his term has "always been my plan," Rubio told The New York Times in an interview.
"I love the work I'm doing," Rubio said. "And plus I always laugh when they say he's going to leave his job. I go, 'Which one?'"
The White House and US State Department also shut down reports that Rubio might be leaving after this week's midterm losses, with both calling it "fake news".
The defence secretary, though, faces a more uncertain political future.
Trump and Vice-President JD Vance have both blamed the Iran war for Republican losses that allowed Democrats full control of the US House of Representatives and the Senate.
In a telephone interview with ABC News, Trump said that he had received "calls about" Hegseth, but added that he had "every intention" of keeping him.
Later on Thursday, he told reporters at the White House that "Pete Hegseth has done a fantastic job", and went on to list what he sees are some of his achievements, including the arrest of Venezuelan leader Nicolás Maduro in January and the Iran war, now into its ninth month.
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The Washington Post reported on Thursday that Trump might be, however, considering ousting General Dan Caine, the chairman of the Joint Chiefs of Staff, or Centcom chief Admiral Brad Cooper. Both played significant roles in offering guidance on the war, which started eight months ago.
Meanwhile, Republican Representative Thomas Massie, a frequent critic of Trump, has said he will push ahead with plans to push for Hegseth's impeachment.
He told news site Politico that he will trigger his resolution to impeach the defence secretary early next week.And on X, he warned that the president should replace Hegseth before a new Democratic-controlled Congress forms in January.
"He should do it while he can still get the replacements confirmed in the Senate," Massie wrote on X.Hegseth has also been rumoured to be interested in running for an open Tennessee Senate seat.
In his video statement, Hegseth denounced the "sham articles of impeachment" effort against him from a "disgruntled, defeated congressman, along with all his Democrat allies", and listed ways the department has fulfilled Trump's America First agenda under his leadership.
That includes, he said, reviving military recruitment, rebuilding the defence industrial base, and rooting out diversity initiatives.
Of the Democrats, he said: "Their policies don't work. So their obsession with political theatre disguised as power drives them to try to destroy their opponents. It's all about the politics of personal destruction."
Vance says Republicans lost big because of Iran war: 'Everyone knows why we lost'
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