His office announced the 71-year-old the South Carolina Republican died on Saturday.
In an interview with Reuters, Trump said he spoke to Graham "minutes" before he was taken ill and ruled out any foul play.
"There are these rumours. But, no, I believe he had a very short-term illness. He had a heart attack," Trump said.
"He was a friend of mine, a great friend of mine, and it's very devastating losing him."
A preliminary finding from the medical examiner of the District of Columbia said the cause of death was "aortic dissection", Graham's office said in an email.
Aortic dissection is a tear in the main artery that carries blood from the heart. The finding showed it was due to arteriosclerotic cardiovascular disease.
Trump has ordered US flags to be lowered in Graham's honour.
The contest to succeed Graham in reliably Republican South Carolina won't impact the broader fight for control of the Senate between Republicans and Democrats in the November midterm elections.
But his death robs Trump of a dependable Senate vote as the president seeks to push his agenda in Congress.
With the ongoing absence of another Republican lawmaker, Senator Mitch McConnell, it effectively narrows the party's Senate majority to a minimal 51 votes until a replacement can be sworn in.
Senate Republicans were already down one vote because of the absence of McConnell, who remains hospitalised recovering from what he described as injuries from a fall at home, as well as mild pneumonia.
Graham died less than two days before the Senate is scheduled to return on Monday from its July 4 break for a compressed work period in which Republicans hope to advance key legislation on defence and national security matters and to confirm Trump nominees including the president's former lawyer, Todd Blanche, as US attorney-general.
Under South Carolina law, the state's Republican Governor Henry McMaster can immediately appoint a temporary replacement to fill the seat for the remainder of Graham's term, which ends in early January.
Republicans hoping to serve in Graham's place for the next full six-year Senate term will compete in a special party primary election on August 11, with a runoff set for August 25 if no candidate wins a majority, according to state law.
The winner will face Democrat Annie Andrews in the November general election.
Graham, a defence hawk, was a prominent supporter of Israel and Ukraine and an opponent of Iran.
Ukrainian President Volodymyr Zelenskiy said he was "deeply saddened" by the news, calling Graham "a true defender of freedom and the values that make our world safer".
On Friday, Graham met Zelenskiy in Kyiv, and the Ukrainian leader said the two discussed Ukraine's air defence needs and a Russia sanctions bill.
US Representative Michael McCaul, a Texas Republican and former chairman of the House Foreign Affairs Committee, said Graham's death could strengthen momentum to pass a bill that would increase US sanctions on Russia - a Graham-championed initiative that won the White House's endorsement last week.
"The best way we can honour Lindsey is to pass his bill," McCaul told Reuters.
Israeli Prime Minister Benjamin Netanyahu said in a statement that Israel had lost one of its greatest supporters.
"I have lost a beloved friend," Netanyahu added.
During the 2016 presidential campaign, in which Graham was among many Republicans who lost the nomination to Trump, he posted on social media: "If we nominate Trump, we will get destroyed ... and we will deserve it." Graham told CNN in 2015 that Trump was "a race-baiting, xenophobic, religious bigot".
Later, after becoming a loyal supporter and frequent golf partner, Graham still publicly disagreed with Trump's decision upon returning to office last year to pardon about 1500 of the president's supporters who attacked the US Capitol on January 6, 2021, saying it could lead to more violence.
"They did not always agree, but ... they figured out how to have a friendship," Republican Senator Tim Scott said.
Graham rose to prominence in Washington in the late 1990s when he was chosen as a manager for the House impeachment case against President Bill Clinton. The House impeached Clinton, but the Senate acquitted him and he remained in office.
In 2018, Graham passionately defended Brett Kavanaugh, who was nominated by Trump for a seat on the US Supreme Court, amid sexual assault allegations against the nominee. Kavanaugh was narrowly confirmed to the high court. Trump, in a CNN interview on Sunday, called it the "finest moment" of Graham's Senate career.
A former Air Force lawyer and member of the South Carolina Air National Guard, Graham was elected to the Senate in 2002, taking over a seat that segregationist Strom Thurmond had held for nearly half a century. Before that, he was elected to the House of Representatives in 1994.
He never married and lived in South Carolina.
