"Mostly, I'm looking for somebody that can interpret the Constitution as written," he said during the exclusive one-on-one interview following the nomination, which aired Sunday morning. "We say it all the time and [Judge Amy Coney Barrett] is very strong on that."
The president also said he thinks Barrett could be confirmed ahead of the Nov. 3 election, saying there's a "tremendous amount of time."
"I think we could have it done easily before the election," he told Hegseth, adding that they were "going to try and have it done quickly."
"I think she'll be confirmed [...] relatively long before the election," Trump said.
Barrett said Saturday in accepting the nomination that she has the same judicial philosophy as the late Justice Antonin Scalia, which is to "apply the law as written."
If Barrett gets confirmed by the Senate, then it will be the president's third nominee since he was inaugurated.
It's not just the Supreme Court on which the president will have a lasting effect, but the lower courts, as well.
"By the end of the term, we'll have almost 300 federal judges and Court of Appeals judges, which is a record. So we will have had a great impact on the court system going forward," Trump said.
Trump has publicly pressured Democratic presidential candidate Joe Biden to release his own list of potential nominees to the Supreme Court. Biden will have a chance to respond to Trump on Tuesday in the first presidential debate.
Fox News' Chris Wallace will moderate the debate, which will cover both candidates' records, the Supreme Court, the coronavirus pandemic, the economy, race and violence in cities, and the integrity of the election.
The president said he has been preparing every day.
"When you're president, you sort of see everything that they're going to be asking. And they may disagree with you, but we've done a great job. We created the greatest economy in history," Trump told Hegseth. "And now it's coming back. We closed it. We saved millions and millions of lives by doing what I did. And now we're bringing it back."
Several Democrats, including House Speaker Nancy Pelosi, have said that Biden shouldn't debate Trump, but Biden has rejected those calls, saying he'll "fact check" Trump himself.