There was no immediate comment from the Vatican as to whether the Argentine-born Francis would accept her request, which was made during a meeting and lunch with Fernandez on the eve of his installation as pope.
But the British Foreign Office made clear that there was no place for Vatican intervention in the dispute.
"The Holy See is clear that it considers the question of the Falkland Islands as a bilateral one between sovereign nations, and that it does not have a role to play. We do not expect that position to change," the Foreign Office said in a statement.
Francis and Fernandez are longtime rivals: As leader of Argentina's Catholics, he had accused her populist government of demagoguery, while she called his position on gay adoptions reminiscent of the Middle Ages and the Inquisition.
But where the Falklands are concerned, Francis has been quoted as saying that Britain "usurped" the remote islands, which Argentina claims and calls the Malvinas.
Argentina and Britain fought a 1982 war over the islands. Earlier this month, the islanders voted overwhelmingly to remain a British Overseas Territory.
Fernandez told journalists Monday after having lunch with the pope that she had asked for Francis' intercession to "facilitate dialogue" with Britain over the islands.
Just last week, British Prime Minister David Cameron said he didn't agree with Francis' views on the Falklands. And on Monday, the Foreign Office recalled the referendum results in its statement, saying: the vote "sent a clear message around the world that the people of the islands want to remain as a British Overseas Territory."
In asking Francis to intervene, Fernandez said she recalled how Pope John Paul II averted war in 1978 between Argentina and Chile over three tiny islands in the Beagle Channel at the southern tip of South America.
With military governments on both sides poised for battle, he sent his personal envoy to mediate the crisis through shuttle diplomacy between Santiago and Buenos Aires, and eventually brought both governments to the Vatican to consider his compromise.
The conflict wasn't entirely resolved until after democracy returned to Argentina, and both sides signed a "treaty of peace and friendship" at the Vatican in 1984, giving the islands to Chile but maritime rights to Argentina.
On Monday, Fernandez gave Francis a picture of a marble monument honoring the 30th anniversary of John Paul II's negotiations, and then used the opportunity to bring up the issue of sovereignty over the Falklands.
They also seemed to have patched up their relationship.
Fernandez gave the new pope a mate gourd and straw, to hold the traditional Argentine tea that Francis loves, and he gave her a kiss.
"Never in my life has a pope kissed me!" Fernandez said afterward.
Fernandez called on the former Archbishop of Buenos Aires at his temporary home, the Vatican hotel on the edge of the Vatican gardens, and the two later had lunch together, a day before she and other world leaders attend his installation Mass in St. Peter's Square that some estimates say could bring 1 million people to Rome.
The Vatican on Monday released details of the Mass, saying it would be a simplified version of the 2005 installation Mass that brought Pope Benedict XVI to the papacy, with many gestures to Eastern rite Catholics and Orthodox Christians in a sign of church unity.
The Vatican also released details of Francis' coat of arms and official ring, both of which are in keeping with his simple style and harking back to popes past: The coat of arms is the same Jesuit-inspired one he used as archbishop of Buenos Aires, while the ring was once offered to Pope Paul VI, who presided over the second half of the Second Vatican Council, the church meetings that modernized the church.
Francis will officially receive the ring and the pallium, a wool stole, during Tuesday's installation Mass, which is drawing six sovereign rulers, 31 heads of state, three princes and 11 heads of government to the Vatican. Fernandez leads the largest delegation with 19 members.
She and her predecessor and late husband, Nestor Kirchner, defied church teaching to push through a series of measures with popular backing in Argentina, including mandatory sex education in schools, free distribution of contraceptives in public hospitals, and the right for transsexuals to change their official identities on demand. Argentina in 2010 became the first Latin American country to legalize same-sex marriages.
According to Francis' authorized biographer, Sergio Rubin, the former Cardinal Jorge Mario Bergoglio was politically wise enough to know the church couldn't win a straight-on fight against gay marriage, so he urged his bishops to lobby for gay civil unions instead. It wasn't until his proposal was shot down by the bishops' conference that he declared what gay activists called a "war of God" on the measure -- and the church lost the issue altogether.
Fernandez issued a perfunctory message of congratulations when Francis was elected last week, calling the election of the first Latin American pope "historic" and saying she hoped that given his namesake, St. Francis of Assisi, the new pope would inspire world leaders to pay greater attention to the poor and pursue dialogue rather than force to resolve disputes.
She has, however, remained unusually silent about the election on her otherwise prolifically active Twitter account, posting a single tweet on his election day: "To your Holiness Francis I" with a link to her letter of congratulations, which wasn't even signed.
Their chilly relations became crystal clear after the Kirchners several years ago stopped attending the church's annual "Te Deum" address challenging society to do better, which is delivered each May 25.
In last year's address, Bergoglio said Argentina was being harmed by demagoguery, totalitarianism, corruption and efforts to secure unlimited power: a strong message in a country whose president has ruled by decree and left scandals unpunished.
The Fernandez meeting isn't the only diplomatic dance Francis will be conducting this week as more than 132 government delegations descend on Rome for the Mass formally installing Francis as the 266th leader of the 1.2-billion strong Catholic Church.
Italian media say Rome civil protection authorities are planning for upward of 1 million people to attend the Mass, numbers not seen since the beatification of Pope John Paul II in 2011, which drew 1.5 million to St. Peter's and the surrounding streets.
One significant VIP is the spiritual leader of the world's Orthodox Christians, Ecumenical Patriarch Bartholomew I. His presence at the installation is the first from the Istanbul-based Patriarchate in nearly 1,000 years since the Great Schism divided the church in 1054.
The Mass will make several gestures toward Eastern rite and Orthodox Christians, with the Gospel being chanted in Greek as opposed to Latin and eastern rite Catholic prelates joining Francis at an initial prayer at the tomb of St. Peter under the basilica's main altar, the Vatican said Monday.
In all, some 33 Christian delegations will be present, as well as representatives of Jewish, Muslim, Buddhist, Sikh and Jain communities. They will see a simplified Mass compared to the 2005 installation of Pope Benedict XVI, with for example fewer cardinals pledging obedience to the new pope.
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