The head of the Catholic Church loves football and owns shares in San Lorenzo, a club in his native Argentina.
The Italian FA said that they have proposed a date of August 14, which has been set aside for international fixtures.
"The date Wednesday August 14 is set aside in the FIFA calendar for internationals and we had already been in negotiations for some time with the Argentine Federation to plan a friendly on neutral turf in Switzerland or Belgium," read a statement.
"Today the FIGC instead proposed Rome as the host city, sending a letter to the general secretary of the Argentine Federation Miguel Silva.
"Alternatively, the Italian Nazionale will use the August 14 date to play a friendly against a different opponent."
Azzurri coach Cesare Prandelli last week suggested a match, with the idea of both teams meeting the Pope in the Vatican beforehand then travelling to the nearby Stadio Olimpico together.