A short period of tighter restrictions - lasting a few weeks - could be announced in the next week, BBC political editor Laura Kuenssberg said.
Schools and most workplaces would be kept open during those weeks.
Health Secretary Matt Hancock has told the BBC the government is "prepared to do what it takes" against Covid-19.
Speaking to BBC Breakfast, the health secretary said there had been an "acceleration" in the number of coronavirus cases in the last couple of weeks, and the number of people admitted to hospital with coronavirus was doubling about every eight days.
He stressed it was "critical" that people follow social distancing guidelines and local lockdown rules - where they apply.
"If we do all these things, we can avoid having to take serious further measures," he told the programme.
'Circuit break'
At a meeting on Wednesday night, the government's chief scientific adviser and medical officer predicted another serious outbreak of the disease.
They forecast that there would be a significant number of deaths by the end of October if there were no further interventions.
The possible measures being discussed - described by the government as a "circuit break" - include asking some hospitality businesses to close, or limiting the opening hours of some pubs and restaurants nationwide.
No final decisions have yet been reached on the next course of action.
Labour's shadow health secretary Jonathan Ashworth has said it is "not surprising national restrictions are back on the table" as the UK's testing system is "collapsing".
The virus is now understood to be doubling every seven to eight days, with more than 3,300 new cases reported on Thursday.
And having been contained to single figures for much of the past six weeks, the daily number of deaths of people within 28 days of a positive coronavirus test has risen above 20 for the past three days.
In France and Spain, a rapid surge has seen both countries record more than 10,000 new positive cases a day, more than 20 times higher the level in June.
It comes as nearly two million people in north-east England are the latest to face local lockdown rules, which came into force on Friday. The restrictions will ban people from meeting other households, and restaurants and pubs will have to shut at 22:00 BST.
New local lockdown rules are also set to be brought in for Lancashire - with the exception of Blackpool - the BBC understands.
The four nations of the UK are all in charge of their own lockdown restrictions, with Scotland, Wales and Northern Ireland implementing slightly different rules to England.
Options for ministers
Under the so-called "circuit break", restrictions could be reintroduced in some public spaces nationwide for a period of a few weeks, but schools and workplaces would be kept open.
One of the ideas suggested by the government's Scientific Advisory Group for Emergencies (Sage) is that some parts of the hospitality sector could be asked to close.
No 10 is also considering the possibility of limiting the opening hours of pubs and restaurants across the country, as has already happened in some areas.
The health secretary said the government's current approach was "targeted interventions" and stressed "a national lockdown is the last line of defence".
"The strategy is to keep the virus down as much as is possible whilst protecting education and the economy," Mr Hancock added.
"And throw everything at the science which eventually is the way we're going to spring out of this."
No 10 is also considering the possibility of limiting the opening hours of pubs and restaurants across the country, as has already happened in some areas.
The health secretary said the government's current approach was "targeted interventions" and stressed "a national lockdown is the last line of defence".
"The strategy is to keep the virus down as much as is possible whilst protecting education and the economy," Mr Hancock added.
"And throw everything at the science which eventually is the way we're going to spring out of this."
Prime Minister Boris Johnson is understood to be deeply reluctant to order another national lockdown, where everyone would be asked to stay at home and businesses to close.
Earlier this week he described the potential impact of a second national lockdown on the economy as "disastrous".
On Thursday morning, Chancellor Rishi Sunak is understood to have presented warnings of the damage to the economy.
And ministers are also concerned about the impact of more restrictions on daily life on those who need treatment for non-Covid related illnesses.
However, the government has not yet taken any final decisions about the next course of action.
It is not yet clear what impact this week's new rule banning social gatherings of more than six people will have on the rate of increase, and No 10 is continuing to monitor the data and take scientific advice.
But it seems increasingly likely that within the next week, the prime minister will tighten the national rules again, our correspondent said.
On Friday, parts of north-east England joined other areas across the UK in being under local lockdown rules.
The measures affect Newcastle, Gateshead, Sunderland, Northumberland, South Tyneside, North Tyneside and the County Durham council area.
As well as the ban on households mixing and early closures for pubs and restaurants, people should also only use public transport for essential travel and care homes are closed to visitors.
An announcement on a possible lockdown in north-west England is expected from the government later on Friday.
The restrictions for Lancashire, but not Blackpool, are expected to include a similar ban on households mixing and a requirement for hospitality businesses to close early.
It is not yet known when the measures will be introduced although the BBC understands it will be imminent.
There are also local lockdown restrictions elsewhere in the UK - including Birmingham, Greater Manchester, Caerphilly, and the Belfast council area.
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