His nomination is the subject of growing scrutiny by members of his own party - including US senators who have the power to confirm or deny his appointment when are asked to vote on it.
"I think some of these articles are very disturbing," Senator Lindsey Graham told CBS on Tuesday. "He obviously has a chance to defend himself here, but some of this stuff is going to be difficult."
DeSantis, who was elected Florida governor in 2018, did not reply to a request for comment. A spokesperson for Trump declined to say whether DeSantis was under consideration for the post.
DeSantis ran against Trump in the Republican primary, and before dropping out, he was considered by some to be "Trump 2.0" - a Republican who could deliver Trump's populist agenda without baggage.
The latest speculation - first reported by the Wall Street Journal - comes as Hegseth meets members of Congress this week to discuss the job and drum up support.
A graduate of Princeton and Harvard universities, Hegseth was an infantry platoon leader in Guantanamo Bay and Iraq, and was awarded the Bronze Star Medal.
In nominating Hegseth, who is also a former Fox News TV host, Trump highlighted the former soldier's education, and his military experience in Afghanistan and Iraq.
"With Pete at the helm, America’s enemies are on notice - our military will be great again, and America will never back down," Trump wrote.
But even as a veteran of the wars in Afghanistan and Iraq, the 44-year-old does not have the extensive experience typical for the cabinet position. He also would be the second-youngest person to serve in the office.
In addition, since his nomination, a police report detailing accusations of an alleged sexual assault in 2017 has surfaced.
The woman quoted in the complaint said that Hegseth, then a Fox host, took her phone and blocked the door to a hotel room while at a Republican conference in California.
Hegseth denies any wrongdoing, saying the encounter was consensual.