The first batch of tests – which give results in half an hour – will be delivered from the Guru Nanak Sikh temple in Sedgley Street, Blakenhall.
People from the community who do not have Covid symptoms are invited to turn up and take a test from 7am to 7pm. Anyone displaying symptoms should not attend.
Those attending the temple must wear a face covering and can be accompanied by one person from their household or support bubble. Results will be provided via text or email.
The scheme aims to identify undiagnosed Covid cases in order to help protect those most at risk from the virus.
It will also provide vital information to help inform the rollout of mass testing technology in the future.
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Councillor Ian Brookfield, the leader of Wolverhampton Council, said: "The first week's 10,000 testing kits will be run from the gurdwara.
"We are interested to see what the take-up is."
The pilot at the temple will be run by trained volunteers and is scheduled to end on November 30.
Mr Brookfield said the council was expecting a second batch of 27,000 kits next week, which are expected to be focused on care homes, hospitals and schools.
For that stage of the pilot, he said, a proportion of the rapid tests are likely to be offered to people who had been in contact with someone who had tested positive, and were "having to self-isolate even though they don't have the virus".
Health Minister Lord Bethell said: “We’ve already come so far since first setting up a national testing programme at an unprecedented pace to help counter Covid-19, but we continue to strive to go further, faster.
“Innovations such as lateral flow tests hold the key to the next phase of our ambition to see mass, rapid testing available to people across the country.
“I’m delighted that the City of Wolverhampton Council and the Guru Nanak Sikh Gurdwara are working with us to pilot the latest technology in Wolverhampton, and I look forward to seeing the fruits of their labour, both in helping target the virus locally, and helping find ways to roll this technology out further soon."
The community-led scheme is separate to the national pilot of rapid testing being rolled out across 67 local authorities, including Wolverhampton, in the coming weeks.
Health chiefs in the city are also looking at the possibility of introducing home testing.