Trump greeted Francis in Sala del Tronetto, the room of the little throne, on the second floor of the Apostolic Palace.
Mrs Trump, who grew up in Slovenia, reportedly speaks as many as six languages acquired during her worldwide modelling career.
Trump's gift for the pontiff was wrapped in a big blue box.
It looks like they posed for a photograph before they split ways, the Trumps to attend a Black Mass, and the pontiff to hear confession.
"We had a fantastic meeting", the president said afterward.
"I signed it personally for you", the Pope offered.More news: Watch The One Love Manchester Concert With Ariana Grande LIVE!
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Tillerson said Trump "hasn't made a final decision" about the US standing in the accord, and likely will not until after he wraps his first overseas trip as president and returns to the U.S.
The Pope gave the president a copy of Laudato Si, an encyclical he wrote on the environment - a pointed gift, given that Mr Trump has threatened to withdraw the U.S. from the 2015 Paris accord on combating climate change.
The CNN story notes that while senior White House officials claimed the Vatican is strict on admittance into its talks, "previous administration officials who helped orchestrate meetings between United States presidents and the Pope said that high-level Catholic staffers who expressed interest in attending the papal sessions were regularly accommodated".
Trump presented the pope with several gifts, including a collection of first editions by Martin Luther King and a bronze sculpture. "I won't forget what you said".
Trump will later meet Italian President Sergio Mattarella and Prime Minister Paolo Gentiloni.
Trump's predecessor, Barack Obama, had a private audience with Francis at the Vatican in 2014 that lasted 50 minutes.