The centre, which monitors all major events, will run several enhanced surveillance activities in all countries where matches are being held throughout the month-long tournament.
Monitoring began on June 4, with no particular concerns reported across Europe in the first week of preparation.
Any urgent threats will be reported as they are identified.
The ECDC estimates 460,000 spectators will watch the games, in stadiums operating at much lower capacity than usual.
Four games will be held at Hampden, with 12,000 fans, around 25 per cent of capacity.
Glasgow is one of the only host cities not to require proof of a negative Covid-19 test, vaccination or a recent infection in order to gain entry. The only other countries which do not require this are Azerbaijan and Spain.
ECDC director Andrea Ammon said “As I am sure many in Europe are looking forward to Euro 2020 and watching their national teams in stadiums across the continent, it is important to remember that the pandemic is not over.
"We must remain vigilant in the face of any rise in infections; ECDC will play its part with increased Epidemic Intelligence activities before, during and after the tournament in an effort to keep everyone safe.”
First Minister Nicola Sturgeon urged fans gathering at Hampden and elsewhere on Monday to be cautious.
“As you enjoy, we hope, lots and lots of Scotland goals, please try and do it safely, for your own sake but for those you love as well,” she said.