Authorities said on Sunday that six cases of the Brazilian variant had been found in the UK, three in England and three in Scotland.
But one of the three infected people in England could not be identified, as they did not complete their test registration card. The government launched an appeal for the person to come forward.
Briefing the House of Commons on Tuesday, Hancock said the search had made significant progress.
“We’ve identified the batch of home test kits in question. Our search has narrowed from the whole country down to 379 households in the southeast of England, and we’re contacting each one,” he said.
A number of potential cases had come forward following the government appeal, he said.
The main opposition Labour Party criticised the government on Monday for failing to identify the case, saying the “failed quarantine system” had put the public at risk.
But Hancock told Parliament that incidents like this “are rare and only occur in around 0.1 percent of tests.”
He said the five other people who tested positive for the Brazilian variant “quarantined at home as they were legally required to do,” and the authorities have been in contact with them.
As a precaution, health authorities have stepped up testing and sequencing in South Gloucestershire, where two of the cases were found.
“We have no information to suggest that the variant has spread further,” Hancock said.
But he said the COVID-19 vaccines currently in use in the UK have not yet been studied against the Brazilian variant.
“We’re working to understand what impact it might have, but we do know that this variant has caused significant challenges in Brazil,” he said.
The Brazilian P.1 variant carries several mutations that may change the behaviour of the virus, said Nick Loman, professor of microbial genomics and bioinformatics at the University of Birmingham.
“These include E484K, which is predicted to make existing antibodies to SARS-CoV-2 less effective, as well as N501Y, which is potentially linked to increase transmissibility,” he said.
Scientists are also concerned that the P.1 variant can re-infect CCP (Chinese Communist Party) virus patients more easily.
“Recent reports from Manaus in Brazil, where the P.1 variant is fuelling a surge in infections, suggest that this variant is responsible for re-infecting individuals who were previously infected with a different variant of the virus,” said Lawrence Young, professor of molecular oncology at Warwick Medical School.