COVID-19 has wreaked havoc globally, with the Serie A campaign suspended in March due to the pandemic.
A decision on the season is expected on Thursday as Serie A looks to follow in the footsteps of Germany's Bundesliga, which resumed on May 15.
"We are working on two possible dates to begin Serie A games, on June 13 or 20," Spadafora told Rai TG3.
"The protocol arrived for resuming the season and it is very similar to the one that was agreed for training. On Thursday, we'll decide if and when to resume."
Defending champions Juventus were a point clear of Lazio through 26 games when the league was halted.
"This emergency has shown some critical issues in the world of football that we will face in a general reform that will happen before the end of the summer," Spadafora said.
"Among those issues we include allowing women's football players to become professional athletes."