UEFA president Aleksander Cerefin says he has a "concrete plan" to end the European season "in August".
BBC Sport reported last month that the Champions League and Europa League are likely to be played that month.
The quarter-finals and semi-finals are expected to be played in two stages.
The head of the governing body of European football told BeIN Sports: "As things are now, I am sure that we can end the European season and that means UEFA competition."
Cerefin said the games "will have to be played without spectators", but he expects the "majority" of domestic leagues to complete the season.
The Premier League has nine rounds of matches remaining, while Paris St-Germain were awarded the Ligue 1 title after the French government canceled the 2019-20 sports season.
The German Bundesliga returned behind closed doors on May 16 and the Spanish La Liga hopes to do the same in June after the players returned to training.
Cerefin said the leagues that do not complete the season "will still have to play qualifiers if they want to participate in UEFA competition" next season.
He added: "Only sporting reasons can be the formula for qualifying for the Champions League and the Europa League."
Euro 2020, which will take place from June 12 to July 12 of this summer, will pass from June 11 to July 11 of next year, after a decision of postpone in March.
The semi-finals and final will be held at London's Wembley Stadium, with 11 other host cities: Amsterdam, Baku, Bilbao, Bucharest, Budapest, Copenhagen, Dublin, Glasgow, Munich, Rome and St. Petersburg.
Cerefin said the idea was to "stay in the same cities" for the tournament in 2021 and UEFA had "talks with nine" of those host cities.
"With three cities, we have some problems. So we will discuss more and, in principle, we will do it in 12 cities. But if not, we are ready to do it in 10, nine or eight," he added.