The health secretary said officials had identified the batch of home testing kits in question, meaning the search "narrowed from whole country down to 379 households in the South East and we're contacting each one".
Mr Hancock said the UK's current vaccines had "not yet been studied against this variant" but that work was underway to "understand what impact it might have".
It is feared that the Manaus variant could spread more easily and be more resilient to vaccines. Scientists say the variant is more transmissible and may be more resistant to existing vaccines than the original virus, and may be able to infect people who have previously had Covid-19.
Two of the three English cases were confirmed in South Gloucestershire but the third could be anywhere in the nation.
The health secretary said there was "no information" to suggest the variant has spread further, but confirmed testing and genome sequencing was being increased in south Gloucestershire "as a precaution". All those aside from the missing case have been told to self-isolate.
"Unfortunately one of these six cases completed a test but didn't successfully complete the contact details".
Around 10,000 Covid tests were delivered across the ME15 postcode in Maidstone
The arrivals came days before the United Kingdom imposed a 10-day hotel quarantine on people arriving from high-risk countries, including Brazil.
"It is important to stress that, as the Secretary of State said, each household will be contacted", a spokesman added. "Incidents like this are rare and only occur in around 0.1% of tests", he added.
However, experts said that the situation in the United Kingdom was very different to that in Brazil and there was still cause for "optimism", not concern.
"There are going to be challenges on the way - and there is always a risk that we might have to go backwards, and that's what nobody wants to do is to actually open up and then have to close down again", he told the BBC.