President Trump praised Senator Lindsey Graham as a “great politician” on Sunday, as he mourned one of his closest friends and allies, whom he said he had spoken to just hours before he died Saturday evening.

In an interview on the NBC show “Meet the Press,” Mr. Trump spoke softly and somberly as he reflected on Mr. Graham’s friendship and political acumen.

“He had a unique ability — he was able to deal with Democrats and Republicans,” he said, adding: “If I had a problem with a Democrat, he could work it out. He was a great politician, actually.”

Mr. Trump said Mr. Graham, a South Carolina Republican, had called him on Saturday night, just hours before the president was notified that Mr. Graham had died. He said that he had spoken with Mr. Graham about the SAVE America Act, the voter identification legislation that Mr. Trump has been pressuring Congress to pass.

Mr. Trump said that Mr. Graham had sounded “perfect” on the call, but he added that the senator, who had landed from a trip to Ukraine hours earlier, had told him that he was “tired.”

Mr. Trump said Mr. Graham was “like a member of the family.” He pointed to his defense of Justice Brett Kavanaugh’s 2018 nomination as part of the Senate Judiciary Committee as “literally, one of the great classics of anyone in the Senate.”

The president said he did not know the cause of Mr. Graham’s death, which the senator’s office attributed to “a brief and sudden illness.” But, in response to a question about reports that the authorities had been called to Mr. Graham’s home to respond to cardiac arrest, Mr. Trump said the account made sense.

“It had to be something like that because he was, other than being tired, he was fine,” Mr. Trump said. “And so it had to be something like that, that would just be a quick end. And maybe that’s not the worst way to go.”

Mr. Trump declined to say who he thought should replace Mr. Graham but added that he was considering “somebody that I think would be great.” South Carolina law permits Gov. Henry McMaster to appoint a replacement to serve out the rest of Mr. Graham’s Senate term and suggests that a special election will be held on Aug. 11.

“I never thought I’d be in this position,” he said, adding that he thought “Lindsey was going to be living forever.”

Mr. Graham developed a reputation as a military hawk who tended to support action against Iran. Last month, after Mr. Trump announced that the United States and Iran had reached a preliminary agreement to halt hostilities, the senator cautiously supported the administration while reiterating the need to be aggressive about Iran.

“We want diplomacy to succeed,” the senator wrote on social media. “But if Iran tries to test us, we will obliterate them.”

After recently returning from a trip to Ukraine, Mr. Graham had been scheduled to appear on “Meet the Press” Sunday morning but was replaced by Mr. Trump and Prime Minister Benjamin Netanyahu of Israel.

During his trip to Ukraine, Mr. Graham had announced that a group of senators had reached an agreement with the White House on imposing sanctions targeting buyers of Russian oil.

The senator had previously backed harsh sanctions on countries that do business with Moscow, though such legislation stalled in deference to Mr. Trump’s push for a cease-fire.

“As Russia intensifies its slaughter of civilians, it is imperative that the legislative and executive branches work together to create tools to exact a heavy price on those who buy Russian oil and natural gas, fueling the Putin war machine,” Mr. Graham and three other senators said in a joint statement on Friday.

President Volodymyr Zelensky of Ukraine said on Friday that he had been grateful to Mr. Graham, posting a photo of the two shaking hands in Ukraine.

Mr. Graham was also part of a group that met with Mr. Zelensky at the NATO summit in Turkey last week.

“Thank you, Lindsey, for recognizing our warriors,” Mr. Zelensky wrote, adding that the senator had briefed him on the legislation. Hours after Mr. Graham’s death, Mr. Zelensky added in another post, “America and the world have lost a determined leader.”