The match, played on Sunday between teams from the city's districts of Neuhof and Hautepierre in a local stadium, attracted some 400 spectators in defiance of rules still in place to battle the virus.
France began easing strict lockdown measures a fortnight ago, but gatherings remain limited to 10 people.
The football game drew widespread condemnation and a promise of “severe sanctions” from city hall official Serge Oehler.
▶ Encore un match de foot qui fait polémique : après Strasbourg dimanche, cette rencontre en région parisienne hier soir a rassemblé quelque… 300 personnes.
— LCI (@LCI) May 27, 2020
But on Tuesday authorities said the priority was health, not punishment, although prosecutors want to track down the organisers.
“Our number one concern is health, it isn't a trap,” said the senior local official Josiane Chevalier, calling for those who attended to be tested.
From Thursday they can go to a testing centre set up at the European Parliament to be screened anonymously, without an appointment nor prior consultation.
“We must absolutely avoid another virus hotspot in Strasbourg and protect the participants” as well as their families, said Chevalier at a video press conference.
“It's important that the participants understand that their behaviour in the next few days can save lives,” added regional health officer Adeline Jenner.
“The test will not be held against them,” she added.
In order to break the chain of transmission that could emerge from the event, Jenner said social distancing measures were essential.
Strasbourg remains in the high-alert “red zone” of domestic virus circulation. Along with Paris, eastern France has been one of the worst hit areas in the country.
So far no-one has been arrested in relation to Sunday's game, but the police are searching for those responsible.
“The objective of the investigation isn't to identify all the spectators, but rather the organisers,” said Strasbourg's chief prosecutor Yolande Renzi, adding that someone must have opened the stadium where the match was held.
Violations of the limit on public gatherings up to 10 people can lead up to a six month prison sentence and a €7,500 fine.
Strasbourg mayor Roland Reis said no one should consider a return game between the two sides.
“If there are other events like those on Sunday, all the work done to contain the virus over the two months of lockdown will be put into question,” he said.
Before the strict lockdown was imposed in March France's football league attempted to keep playing behind closed doors, but this was abandoned after dozens of Paris Saint Germain fans turned up outside the stadium where their team was playing.
Both the country's football and rugby leagues have formally ended the 2019/2020 season and professional matches are not expected to restart until September.