It would mark the first time a pope has visited the predominantly Catholic island of 350,000 inhabitants.
The Argentine pope, who will turn 88 next month, could go to Ajaccio to participate in a conference on popular devotion in the Mediterranean scheduled for December 14 and 15.
The Vatican has said the project is "under study", without specifying a date.
Neither the French government nor the embassy of the Holy See in Paris would comment when contacted by AFP.
Rumours of a papal visit have already sent hotel reservations in Ajaccio peaking for the mid-December weekend.
The head of the Catholic Church has been to France twice since becoming pope in 2013, once to Strasbourg in 2014 and last year to Marseille.
The conference in Ajaccio, which is expected to include clergy and academics from around the Mediterranean, comes a week after the highly anticipated reopening of Notre Dame Cathedral in Paris -- for which Francis declined an invitation from President Emmanuel Macron.
Located 160 kilometres (100 miles) south of the French mainland, Corsica is the fourth-largest island in the Mediterranean. Around 90 percent of its population is Catholic, according to the local Church, and religious traditions such as processions remain deeply rooted.
The Holy See and Corsica also share historical ties, such as the establishment of monasticism under Gregory I in the sixth century or the creation of the Papal Corsican Guard in the 17th century.
From Malta to Sicily via the Greek island of Lesbos, Francis has widely travelled the Mediterranean, a region that encapsulates various priorities of his pontificate, such as interreligious dialogue and the welcome of migrants.