Initial reports suggest the father, Stéphane Saada, 50, shot his wife Corinne, 51, and their two children before turning the gun on himself.
The family was visiting from the south-east city of Grenoble. An investigation has been launched by local police with the involvement of forces in Grenoble and Paris.
Events took place in Algajola, a popular coastal resort between the towns of l'île-Rousse and Calvi. The date of the deaths has not yet been confirmed but the bodies were "in an advanced state of decomposition," reported Le Figaro.
The newspaper also quoted a source close to the inquiry who said the man had sent a letter to local authorities "announcing that he planned to kill himself and his family."
At a press conference on Tuesday the local prosecutor confirmed that the father was in an "advanced state of depression" and had "personal and professional problems."
Post-mortems on the bodies will be carried out over the next two days.