The World Health Organisation on Tuesday said it was concerned about the easing of Covid-19 restrictions by nations hosting Euro 2020 matches.
Robb Butler, an executive director at the WHO's European regional office, said some of those countries were already experiencing rising numbers of cases.
"A few of the stadiums hosting the tournament are now increasing the number of spectators allowed," Mr Butler said.
The UN health agency did not single out any cities, but Britain said on Tuesday that more than 60,000 spectators would be allowed at Wembley stadium in London for the semi-finals and final of the tournament.
Originally, it was going to limit the crowd to 40,000.
European football association Uefa has been in talks with the UK government to ease travel restrictions, allowing as many as 2,500 VIPs to attend the final on July 11.
In some "host cities, Covid-19 cases are already on the rise in the area where matches will be held", Mr Butler said.
In areas where infection rates are increasing, WHO Europe called on the cities concerned to act quickly.
"Learning from experience, we must act fast on signals showing increasing cases," Mr Butler said.
"Expanding testing and sequencing; stepping up contact tracing; and building very high vaccine uptake fast among those vulnerable and most at risk."
In Denmark, 29 cases have been detected in connection with the Euro games being played in Copenhagen.
The infections involved people who were already ill during the match or were infected during the game, Anette Lykke Petri, a health authority official, said on Tuesday.
"In theory, there could be more people infected," Ms Petri said.
In Denmark, the permitted audience has also recently been increased to 25,000, from 16,000.
The higher number was first applied for last Thursday's game between Denmark and Belgium.
In Budapest, games at the Puskas Arena, which has a capacity of 68,000, have been played to a full stadium.
Uefa stripped Dublin and Bilbao of hosting roles for the tournament because the capacities allowed were too low.
Among the host cities, Russia's St Petersburg in particular has reported an increase in infections in recent days.
Case numbers are decreasing in Spain's Seville and in Rome, Italian authorities assure that no cases of Covid-19 linked to the competition have been detected.
While the situation across Europe has improved over the past two months, the WHO has urged continued caution.
"Although we have come far, we have not come far enough," Hans Kluge, the WHO's regional director for Europe, warned in early June.
Vaccinations were still too low to protect the region from a resurgence, Mr Kluge said.