Leicester could be kept in the current stage of lockdown for another two weeks, because of a local surge in coronavirus cases.
That means pubs and restaurants would remain closed when they open in the rest of England from Saturday, and social distancing guidelines would stay at 2m (6ft).
But local politicians say it is unclear how restrictions can still be maintained in one area.
What's meant by 'local lockdown'?
Restrictions, or lockdowns, could be used in specific areas in England to counter localised coronavirus "flare-ups".
The Joint Biosecurity Centre, which advises the government on the Covid alert level, will work with local health teams "to address local spikes in infections".
It is unclear whether a lockdown would focus only on the exact location of an outbreak - like a school, care home or workplace - or would cover a whole area, such as a postcode, a town, or even a city.
Councils already have the power to lock down a premises if it presents a health hazard. But the government has also hinted at the idea of locking down a whole city.
However, local politicians say they do not have the powers to enforce such measures, and have asked for more clarity.
A Department of Health spokeswoman said further guidance on containing such outbreaks would be published "shortly".
What could happen?
Recent outbreaks have occurred in the UK:
At a meat factory in Cleckheaton, West Yorkshire At another meat factory in Anglesey in Wales In the city of LeicesterThe meat factories were closed, with all staff told to self-isolate and get tested. In Cleckheaton, those deemed healthy enough were later allowed to return.
Councils say the plan would be similar for schools. If enough cases were detected, a school could be closed and all staff and pupils could self-isolate for 14 days.
Risk assessments could be carried out afterwards to decide when a premises might be reopened.
Public health officials have also suggested people in affected areas might receive council leaflets or be visited by local health workers or politicians to spread awareness.
Local lockdowns could also mean more widespread testing, according to Krishna Ramkhelawon, director of public health for Southend-on-Sea Council: "The most important thing would be to define the scope of an outbreak, and possibly test everyone in an area."
What's happening in Leicester?
The city has seen a recent rise in positive cases of coronavirus.
In the two weeks to 23 June, there were 866 positive cases in the city - 29% of the 2,987 who have tested positive since the pandemic began.
The cases are thought to be mostly among younger, working-age people living in the east of the city.
The government is now recommending maintaining the current lockdown for two weeks beyond 4 July.
However, the city's mayor has said local officials still do not have the information they need to maintain the lockdown in their area.
The Department of Health said there are now four mobile testing units in the city and thousands of home testing kits have been made available.
Image copyright PA Media Image caption Officials say buildings where lots of people congregate would be more likely to be outbreak hotspotsCould a whole area be locked down?
Local councils say they do not currently have the power to do this.
Mayor of Greater Manchester, Andy Burnham, has said he "struggles to see how it can be effectively enforced".
He added that singling out a particular area for restrictions would also cause tensions.
Short of barricading the streets, councils say they are not sure how they could enforce heavy restrictions such as travel bans in any single place.
Have other countries tried local lockdowns?
In Germany, local authorities have the power to vary the level of restrictions in individual states, and a number of small lockdowns have been imposed recently.
On 23 June the state of North Rhine-Westphalia reimposed lockdown measures in two districts, after a spike in infections connected to a meat processing plant. Bars, museums, cinemas and gyms were closed, stricter social distancing was reintroduced, and local residents were asked "not to travel to other districts".
However, there has been resistance to this approach. In Gottingen, police had to enforce quarantine on a tower block where 120 residents had tested positive.
Meanwhile in China, following a recent outbreak in Beijing, a local lockdown has been enforced in parts of the city.
People in medium or high-risk areas cannot leave their neighbourhood, rail and flight services have been cut, and schools, swimming pools and gyms closed.
What has been said about the rest of the UK?
Public Health Wales said that a local lockdown was under consideration after the recent outbreak in Anglesey. However, First Minister Mark Drakeford said that any decision would be not be taken lightly.
The Scottish government - for which public health teams work for the NHS, rather than councils - says it is developing a "responsive system of community surveillance" at a national, regional and local level to identify outbreaks quickly.
In Northern Ireland, the government says that any potential clusters or outbreaks will be handled using "appropriate infection control" in line with its normal guidelines for handling an outbreak of a disease.
Additional Reporting by: Ben Butcher and the Local Democracy Reporter Service