In an audio he can be heard saying: “I’d like to be Pope, that would be my number one choice!” – before saying he has “no preference”.
Known as the “conclave”, the process will start once the nine day mourning period for the late Pontiff ends.
On Saturday, Trump attended the funeral of the former Pope Francis, who died on Easter Monday aged 88.
The Vatican says the conclave to elect the new Pope will begin next Wednesday.
135 Cardinals under the age of 80 from across the world will begin a series of secretive votes to find a successor to Pope Francis.
Preparations include installing the chimney where white smoke will rise when his successor is chosen.
Pope Francis chose his place of burial in St Mary Major Basilica, near an icon of the Madonna that he revered, because it reflects his “humble, simple and essential” life, the archbishop who administers the basilica said on Friday.
The 88-year-old was hailed for his leadership of the Church, proving popular among young people and known for his outreach to those most in need in society. The Vatican said 200,000 people attended the funeral.
Presidents, prime ministers and cardinals were among those bidding a final farewell to the religious leader on Saturday.
LBC’s Shivani Sharma speaks to pilgrims in Rome about Pope Francis’ lasting legacy.
US President Donald Trump joined UK Prime Minister Sir Keir Starmer and a host of other world leaders and dignitaries for the open-air funeral mass in St Peter’s Square.
Volodymyr Zelenskyy appeared to receive a round of applause as he stepped out of St Peter’s Basilica.
Mr Trump and Mr Zelenskyy met before the funeral and spoke about the war in Ukraine.White House communications director Steven Cheung said the pair “had a very productive discussion”.
Mr Zelenskyy’s spokesman, Serhii Nykyforov, said the meeting lasted around 15 minutes.
A photograph of the two leaders shows them sitting opposite one another on chairs within St Peter’s Basilica, after both had paid their respects in front of the pontiff’s coffin.
The face-to-face meeting is the pair’s first since their tempestuous clash in the White House at the end of February.
Earlier, the Italian cardinal at the heart of the Vatican’s “trial of the century” has announced that he is withdrawing from participating in the upcoming conclave to elect a new pope for “the good of the church”.
Cardinal Angelo Becciu’s status has dominated discussions in the days after Pope Francis’s death amid questions about whether he would participate in the conclave to elect his successor.
After his 2020 downfall, Mr Becciu had said he would not participate in any future conclave.
But in recent days he had asserted he had a right to enter the Sistine Chapel with other cardinals on May 7.