The UK is now “seeing a second wave” of Covid-19, Prime Minister Boris Johnson said, adding: “It was inevitable that we would see it in this country.”
Mr Johnson said he did not “want to go into larger lockdown measures” but tighter social distancing restrictions may be needed.
Previously, new rules had been confirmed for Lancashire, Merseyside, parts of the Midlands and West Yorkshire.
It also comes when the UK recorded 4,322 more confirmed cases of Covid-19.
“Obviously, when you look at what’s going on, you wonder if we need to go further than the rule of six that we introduced,” the prime minister said, referring to the new measures introduced on Monday.
He added that “the British have done an incredible job – they have lowered this peak by discipline” but that “people are struggling to maintain that, it is difficult to maintain this discipline for a long time”.
Speaking in Oxfordshire, Mr Johnson continued: ‘As we look at this particular curve and what’s going on right now, we’re clearly going to keep everything under review. I don’t want to go into a second nationwide lockdown at all, it’s the last thing everyone wants.
“I don’t want to go through any bigger lockdowns at all, we want to keep the schools open and it’s fantastic that the schools have come back the way they did. We want to keep the economy open as far as possible, we want to make businesses work.
“The only way to do it is, of course, if people follow the instructions. “
Earlier, BBC political editor Laura Kuenssberg said the government was considering a short period of stricter rules which could be announced next week.
Described by the government as a “cut”, the measures envisaged could involve the reintroduction of restrictions in public spaces for a period of a few weeks. Schools and workplaces would remain open.
Ideas suggested by the government’s Scientific Advisory Group for Emergencies (Sage) include shutting down parts of the hospitality industry.
Meanwhile, new measures to curb the spread of the virus have been introduced in parts of the North West, Midlands and West Yorkshire.
This includes Lancashire (except Blackpool), Merseyside and the boroughs of Warrington and Halton in Cheshire.
New restrictions are also put in place in Wolverhampton, Oadby and Wigston in Leicestershire, and in all parts of Bradford, Kirklees and Calderdale.
Some areas of West Yorkshire had already been subject to restrictions put in place in early August but they have been relaxed. They are now subject to these new rules.
In total, nearly 4.7 million people will be affected by the new restrictions, which prohibit separated households from meeting at home or in private gardens.