Like most leagues around Europe, Serie A was suspended in March because of the Coronavirus pandemic.
Serie A clubs have already been cleared to resume group training.
Most of the top flight clubs previously voted to restart the season on June 13, but that date will be used to clear Coppa Italia second leg semi-final games, with the league resuming a week later.
Juventus will travel to face AC Milan after the first leg ended 1-1 in February while Inter Milan trail Napoli by a goal.
"Italy is starting up again and it is only right that football does, too," Spadafora said on Thursday.
"The league resumes on June 20 - I have already consulted with prime minister Conte. I hope we can complete the Coppa Italia in the week from June 13 to 20. It would be a signal to the benefit of all Italians, given it's the competition shown on public service [television].
"My wish is coming true for the whole world of sport to start again at the same time. We deny the story that I had any reservations about the world of football."
Juventus were leading Lazio by a point when the league was suspended.
The German Bundesliga became the first major European league to resume on 16 May while the Premier League has outlined plans to restart on June 17.
The French Ligue 1 has however cancelled its season, with Paris Saint Germain declared champions.
