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Graham's spokesperson said the death certificate would be pending until "all the toxicological and microscopic testing" was finalised.
The senator had just returned from a trip to Kyiv, Ukraine, where he met with President Volodymyr Zelensky on Friday.
Zelensky said he was "deeply saddened" by Graham's death, writing on X: "Lindsey was a true defender of freedom and the values that make our world safer."
Graham was known for his interventionist stance on foreign policy, and had strongly supported the war in Iran.
Last month, he told CBS that the US would "obliterate" Iran if it did not submit to US control of the Strait of Hormuz.
Israel's Prime Minister Benjamin Netanyahu said on Sunday, "Lindsey understood that the security of Israel and America are inseparable."
Israel lost "one of its greatest friends", he added.
Graham had previously been a critic of Trump, and ahead of the 2016 presidential election said: "If we nominate Trump, we will get destroyed... and we will deserve it."
However, he voted against convicting Trump in his 2021 impeachment trial, and supported Trump's election in 2024.
"There is a dark side to Donald Trump... and he was a very good president. But I am sticking with him because I saw what he did," Graham told the BBC in 2023.
Graham cited Trump's record on the US southern border, the killing of Iran's powerful military commander Qasem Soleimani and the appointment of conservative judges.
Following Graham's death, South Carolina Governor Henry McMaster has the power to appoint a temporary replacement to serve out the remainder of Graham's term, which ends in January.
His successor will be chosen in the November midterm elections.
Before Graham's death, Republicans held a 53-47 majority over Democrats in the Senate. Both parties are fighting for Senate control in November.
