Coronavirus outbreaks have been reported in Leicester and at a meat processing plant in Yorkshire.
About 25% of the city’s 2,494 confirmed Covid-19 cases were reported in the last two weeks, according to official figures.
Health Secretary Matt Hancock said officials were working hard to track those affected.
Meanwhile a number of confirmed cases have reportedly been linked to the food factory in Cleckheaton.
Leicester City Council’s public health director said the numbers were “relatively small” but of concern.
Ivan Browne said the city had high levels of health conditions such as diabetes, “pockets of deprivation, and a very significant Black, Asian and Minority Ethnic population”.
“We know that these factors combine to create a high-risk, more vulnerable population that’s more susceptible to coronavirus,” he said.
Despite the relaxing of lockdown rules, Mr Browne warned that the country was still in the grip of a pandemic.
At the daily Downing Street briefing, Mr Hancock said there was an “outbreak right now in parts of Leicester” and that officials were working hard to track those affected.
He also mentioned a further clutch of cases in Kirklees, West Yorkshire.
Huddersfield Labour MP Barry Sheerman said the cases were apparently associated with a meat processing plant in Cleckheaton.
In a tweet, Kirklees Council’s public health director Rachel Spencer-Henshall said there had been a number of positive cases of Covid-19 at a workplace in the area.
The authority had provided support to managers and employees to minimise further transmission, she said, and had offered testing to staff.
Mr Hancock praised local authorities and public health officials for doing a “magnificent job”.
Earlier, it emerged workers at two food processing plants in Wales had tested positive for coronavirus.