It means the ticket demand is already oversubscribed by almost four times, despite having only been open for a month.
"Yesterday (Friday) we were at 3.7 million requests for one million tickets, so it's a much bigger demand than the offer," said Jacques Lambert, president of the Euro 2016 tournament's organising committee.
Lambert said most requests had been for the cheapest tickets and fans could increase their chances of getting tickets by applying for more expensive ones.
Requests have come mostly from France (55 per cent) but also a lot from around Europe and 200 countries in total.
The first phase of the ticket sale is open until 10 July and requests can only be made on the euro2016.fr website.
In total, 2.5 million tickets will be sold for the 51 matches in the new expanded 24-team tournament.
While many of those tickets are reserved for sponsors, partners and others, 1.8 million tickets will go to the general public.
Each person can apply for up to four tickets.
