"The Euros in 2020 could be held all over Europe," Platini told a news conference.
"You could have one country with 12 host cities, or we could have it in 12 or 13 cities all over Europe. It is just an idea, but in these days of cheap air travel anything is possible.
"The majority of the executive committee thinks it's a good idea and says we should really have a good think about this.
"We are going to have a meeting with all of our national federations coming up to December and in Decemeber or January we're are going to make a decision."
Turkey had been favourite to host the event in eight years' time but their bid has been complicated by Istanbul's campaign to also host the Olympic Games in the same year.
Platini said the idea for a multi-country Euros came to him "a while ago" and had been met with interest from his executive.
"It's an idea I feel really passionate about, it will be a lot easier from a financial perspective for all the countries.
"If you need to build airports or 10 stadiums in a country, this would be rather easy because it would be one stadium per host city. That's all we can say at the moment.
"We have not decided anything yet, but by January we will decide if we are going to have the Euros in one country or all over Europe."
UEFA originally set a deadline of mid-May for hosts to express an interest, but although they said two or three countries other than Turkey had done so, they did not name them publicly.
One reason for UEFA not making them public is because clearly they were not satisfied the interested countries could stage the event, hence Platini's idea for a multi-country tournament.
Platini said during the UEFA Congress in Istanbul recently that he would support a Turkey bid, but has now retreated somewhat from that stance, saying on Saturday: "We are not going to wait until we know whether Turkey are going to get the Olympics.
"It creates a problem for us. We do have other candidates.
"Everyone has the possibility to host it."
When asked how fans would be expected to attend the matches if they wanted to travel around Europe, he replied that in this day of cheap air travel, movement around Europe was not as prohibitive as it once was.
"And it is easier to go from London to Paris or Berlin than Cardiff to Gdansk," the Frenchman said.
"It would be four games per venue. It is a great debate."
Platini's plan is not without merit. It means that matches could be staged in a number of cities around the continent, in a way, bringing the Euros to the fans, rather than the other way round.
The singlular, or co-hosting identity of the tournament might be lost, but fans in some unlikely places could see matches in their home cities.
The next Euros, in France in 2016, will comprise 24 teams, but Platini said it was unlikel
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