A source told the newspaper that Hancock is considering “all options” to control the thrust.
This comes after food manufacturer Samworth Brothers confirmed that a “handful” of employees at its Leicester sandwich factory had contracted the deadly virus.
The Sun said more cases were reported in a Sainsbury supermarket, while five schools have also closed.
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Andrew Bridgen, the Conservative MP for North West Leicestershire, said it was “inevitable” for the government to look for ways to slow the spread of the virus in Leicester.
“Unfortunately, the fact that the government must consider specific measures to control the spread of the virus in Leicester is now inevitable in order to protect the rest of the country,” he said.
However, a source told the Sunday Times that a decision has yet to be made.
“So far, local measures have been taken to lock down hospitals and medical offices where an epidemic has broken out, but this is more serious than that, although no decision has been made yet,” said the source.
This comes as the number of coronavirus deaths in the UK increased by 100 yesterday – the smallest increase in foreclosure on Saturday.
It has brought the total number of deaths in the UK by Covid-19 reported by the Department of Health and Welfare to 43,514.
On July 4, Boris Johnson is preparing to reopen the hospitality industry, with pubs, restaurants and bars among the businesses that will reopen for the first time in almost four months.
It is believed that the decision to put in place a local lock will be made by Public Health England, the Biosafety Center and local directors of public health.