Milei, a maverick right-wing libertarian, in the past called Francis an "imbecile", a "son-of-a-bitch preaching communism" and the devil's man on earth - but has softened his tone since taking office in December.
The pope, for his part, has said he did not pay too much attention to the insults, telling Mexican broadcaster N+ that what matters is what politicians do in office, rather than on the campaign trail.
Francis is due to lead a canonisation Mass for Maria Antonia de Paz y Figueroa, better known as "Mama Antula", an 18th century woman who renounced her family's riches to focus on charity and Jesuit spiritual exercises.
The ceremony comes as Argentina is facing its worst economic crisis in decades, with inflation at more than 200%. Francis has said he hopes to be able make his first trip back to his homeland since being pope in 2013 in the second half of this year.
The Argentine leader, who has said he may convert to Judaism, will attend the service in St Peter's Basilica with his entourage, and is set to have a private audience with Francis on Monday.
Mama Antula was the daughter of a wealthy landholder and slave owner.
She promoted the spiritual exercises -- a mix of prayers and meditations -- walking thousands of kilometres barefoot, involving the rich and poor, despite the Jesuits having at the time been banished from Latin America.
Francis, himself a Jesuit, described her on Friday in a meeting with pilgrims from Argentina as a "gift to the Argentine people and also to the entire Church," praising her dedication to the poor.
Quoting from his past writings, the pope condemned the "radical individualism" that permeates today's society as a "virus", in words that may jar with Milei's radical free-market instincts.
On Monday, wrapping up a week-long overseas tour that took him to Israel before Italy and the Vatican, Milei is also due to meet Italy's President Sergio Mattarella and Prime Minister Giorgia Meloni.
(Additional reporting by Lucinda Alice Elliot in Buenos Aires, editing by Clelia Oziel)
By Alvise Armellini