The Argentinian pontiff was handed the sign at a general audience on Wednesday, with a photograph of the incident surfacing on Twitter shortly afterward. The image was then retweeted by Argentina's President Fernández de Kirchner.
Gustavo Hoyo, who gave the sign to the Pope, told Argentina's Clarin newspaper that "when he passed by, I explained what this was about and he kindly took the placard and got the picture taken".
"He could have chosen not to do it, but he did," added Mr Hoyo.
The placard read: "It's time for dialogue between Argentina and the United Kingdom over the Falkland Islands." It bore a map of the islands, called the Malvinas in Argentina.
#MalvinasArgentinas @Pontifex_es pic.twitter.com/vxewfUd1rS
— Cristina Kirchner (@CFKArgentina) August 19, 2015Reports have suggested the Vatican maintains Pope Francis was a victim of entrapment by Argentine activists and the pontiff didn't know what he was holding.
"The picture does not indicate the Pope has taken a position on the issue," said Father Ciro Benedettini, a Vatican spokesman.
In a written statement he added that the sign, "calling for dialogue between Argentina and the United Kingdom, was taken at a general audience, where many faithful give various objects to the Pope, often to get themselves photographed".
Argentina invaded the Falklands, 300 miles off its coast, in 1982 to claim the British overseas territory.
Then Prime Minister Margaret Thatcher ordered British forces to retake the islands, and 900 lives were lost.
In a referendum held in 2013 the islanders overwhelmingly voted to remain British.
At a memorial mass in 2012 marking 30 years since the Falklands war, the then Cardinal Bergoglio, Archbishop of Buenos Aires said: "We come to pray for those who have fallen, sons of the homeland who set out to defend their mother, the homeland, to claim the country that is theirs and they were usurped."
