Serie A has been suspended since March 9 due to the coronavirus pandemic in Italy where over 15,000 have died - almost a quarter of the global death total.
Several club presidents have suggested that the season should be called off, with Lazio and Juventus fighting it out for the Scudetto.
But FIGC president Gabriele Gravina believes the season must be finished, even if it runs into the 2020/21 campaign, and fears they could face legal challenges if they do not.
"The only serious way to handle a serious emergency of this kind is to complete the 2019-2020 season competitions this year," Gravina told state broadcaster RAI's Domenica Sportiva programme.
"We are working on a whole series of possible solutions to manage the situation in the best possible way."
"Could it finish in September or October? That is one hypothesis. It's a way to avoid jeopardising not only the 2019-20 season but also the 2020-21 season."
Gravina explained that if the current season was abandoned, Italian football could suffer "an avalanche of litigation to which we would be subjected by all those who could be considered damaged in their rights and interests."
This could include teams who were relegated and those who missed out on European football next season.
He added: "We would really run the risk of the championship taking place only in the courtrooms."
Additional reporting from Reuters.
