Russia was suspended from world athletics in November after an international investigation uncovered damning evidence of widespread doping and corruption.
More than 20 Russian track and field athletes have also tested positive in re-tests of samples from the Beijing and London Olympics, putting their participation in this summer's Rio games in doubt.
"We are aware of what happened in Russia. We have been paying special attention on the Russian team," UEFA's head of medical and anti-doping Marc Vouillamoz told reporters at the headquarters of the French anti-doping agency, which will run the testing for the tournament.
Vouillamoz said UEFA had been working closely with the United Kingdom doping agency (UKAD) on the issue and that the Russian team and a number of the squad playing at home and overseas had been tested in recent months.
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UKAD has taken charge of testing Russian athletes while the local agency is suspended for non-compliance following a string of doping scandals.
Without giving details, Vouillamoz said some of the 24 national teams competing in the tournament, which begins on Friday, had already been tested two or three times more than others. Russia kicks off its campaign against England on June 11.
Although, on the surface, doping has not been a major problem in football as it has been for sports like athletics and cycling, there have been recent instances of players using illegal substances to gain an unfair advantage.
Dinamo Zagreb midfielder Arijan Ademi was banned for four years after he failed a drug test following a Champions League match in September. France international Mamadou Sakho was given a provisional suspension in April for a doping offence.
Since 2004, only Croatia striker Ivica Olic has failed a test during the tournament, but was cleared to continue playing.
"I don't have an explanation (on why so few positive tests), but the results are comparable to other team sports," Vouillamoz said.
In the month running up to the competition, UEFA has conducted over 1,000 tests, including urine, serum and blood passport samples. That adds to almost 1,300 since the start of the year and a further 874 in European club competitions.
For each game, two players from each team will be tested with the possibility of random testing on non-match days.
Closer cooperation and data sharing between UEFA and national federations, which carry out the bulk of tests, are also now in place, making the anti-doping programme for the tournament "the most extensive" to date, Vouillamoz said.
UEFA also hopes that with samples now kept in storage for four years, and possibly 10 years soon, it will put off potential offenders, who could face retrospective sanctions.
"This is obviously a very (strong) deterrent message and a factor for players and teams to keep in mind because this could happen with UEFA competitions," he said.
(Reporting by John Irish; Editing by Catherine Evans)
