On Tuesday, UEFA will hold a video conference with delegates from its 55 member nations to discuss how and when football will be able to proceed.
One option which has been mooted is to delay Euro 2020, either until later in the year or to 2021, to allow the domestic seasons to be completed.
Italy has been the worse-hit country in Europe by the virus, with the country on lockdown until at least April 3, and FIGC boss Gravina has confirmed he wants Euro 2020 to be postponed.
"We will try to reach the end of this [Serie A] season because it is fair and correct to have an outcome for the many efforts and sacrifices from our clubs," Gravina told Mediaset on Sunday.
"The hope is that this happens by June 30, without forgetting that in addition to Serie A there are other championships that must be resolved, and we must also include the Champions League and the Europa League. The deadline is June 30, eventually we will see if we have to go beyond."
Regarding European competition, Gravina said: "On Tuesday we will tackle this issue; the primary principle is the protection of health. Italy is currently two weeks ahead, the other [countries] probably do not yet know the exact size of things.
"We'll ask UEFA to postpone the playing of the European Championship."
Juventus lead the way by one point in Serie A, with Lazio in second, and both have 12 Serie A games remaining. Antonio Conte's Inter are nine points adrift of the leaders in third place, with a game in hand.