Restrictions will get tougher if rules are not followed, Matt Hancock says, as £10,000 fines for those who fail to self-isolate are introduced in England.
The health secretary told the BBC's Andrew Marr show the country was facing a "tipping point and we have a choice".
The prime minister is understood to be considering a ban on households mixing, and reducing opening hours for pubs.
On Monday there will be a briefing on the latest coronavirus data led by two of the government's leading scientists.
In the broadcast, at about 11:00 BST on Monday, Prof Chris Whitty, the UK government's chief medical adviser, with Sir Patrick Vallance, the government's chief scientific adviser, is expected to say the trend in the UK "is heading in the wrong direction" with the country at a "critical point in the pandemic".
"We are looking at the data to see how to manage the spread of the virus ahead of a very challenging winter period," Prof Whitty is due to say.
Earlier, asked if England could face another national lockdown, Mr Hancock said: "I don't rule it out, I don't want to see it."
"If everybody follows the rules then we can avoid further national lockdown," he said.
In the BBC interview, he also:
- Said he would call the police on people who refused to self-isolate
- Denied the government was overreacting given deaths and hospital admissions remain relatively low
- Said there was still hope a vaccine would get "over the line" this year
The move under consideration by PM Boris Johnson could take the form of a two-week mini lockdown in England - being referred to as a "circuit breaker" - in an aim to stem a recent surge in cases.
On Sunday, a further 3,899 new Covid-19 cases and 18 deaths were reported in the UK.
Meanwhile, visitors have flocked to Blackpool this weekend, despite police warning against having a "last blast" in the resort before tighter restrictions come into force in the rest of Lancashire on Tuesday.
Elsewhere, London Mayor Sadiq Khan has said he will meet council leaders on Monday and then recommend any London-specific measures to ministers. He believes the capital city may be just "two or three days" behind the hotspots of the north-west and north-east of England.
People in England who refuse an order to self-isolate could be fined up to £10,000 from 28 September.
The new legal duty requires people to self-isolate if they test positive for coronavirus, or are traced as a close contact.
New measures also include a one-off £500 support payment for those on lower incomes, and a penalty for employers who punish those told to self-isolate.
Fines will initially start at £1,000 rising to £10,000 for repeat offenders, and for "the most egregious breaches".
Until now, advice to self-isolate has been guidance only.
More than 19,000 fines have been issued in England and Wales for alleged breaches of coronavirus laws, the attorney general said earlier this week, but more than half have not been paid so far.
The idea of introducing fines sounds good on paper, given it is believed only one in five people fully self-isolate when they need to.
But there is real concern it will have an unintended consequence - and end up discouraging people from having a test in the first place or taking calls from contact tracers.
Self-isolation can be costly for some in terms of lost income and job security (even with the offer of £500 payments).
There is now growing concern among experts - both those advising government and those looking in from the outside - about the path being taken by ministers.
First of all, the messaging is confusing.
One minute the public is being told to Eat Out to Help Out and to get back into the office, then they are being told to curb their mixing and not to gather in groups of more than six.
There is also a growing sense that the public is tiring of the battle to contain the virus with ever more draconian messages.
The idea of curbing the spread of the virus to stop the NHS being overwhelmed brought people together in the spring.
But what is the aim now the NHS was not overwhelmed? Suppression of a virus that clearly can't be suppressed without a huge cost to society?
As Prof Robert Dingwall, a government adviser, put it last week, this virus is here "forever and a day" and the public may just be growing comfortable with the idea that people will die - just as they accept that people die of flu ever year.
Earlier this week, Boris Johnson told the Sun he had "never much been in favour of sneak culture, myself" and people should speak to Covid rule-breakers before reporting them to the police.
In contrast, Mr Hancock said he would call the police on his neighbour if they were breaking rules, saying it was "absolutely necessary" to break the chains of transmission.
Asked whether the government's response was an overreaction given coronavirus death rates were still low, Mr Hancock said the number of hospital admissions was rising and an increase in deaths would follow.
He said he was "very worried" about the latest data which suggested the UK could be on the same path as Spain and France - where deaths and hospitalisations are increasing - without effective action.
"We have seen in other countries when the case rate shoots up, the next thing that happens is the numbers going into hospital shoot up," he said.
"Sadly, we have seen that rise, it is doubling every eight days or so - people going into hospital - then, with a lag, you see the number of people dying sadly rise."
He said it was worth comparing the UK to Spain - where the number of cases have been increasing - and Belgium - which appears to have reversed an upward trend - saying "one gives the warning, the other gives us hope".
Currently, large swathes of the UK, where cases have spiked, are living under tighter local restrictions. London Mayor Sadiq Khan has warned London could be next.
Mr Hancock, who has spoken to the mayor this weekend, told Times Radio he would not rule out the possibility that Londoners could be told this week to avoid the commute and get back to working from home.
Mr Hancock said he remained hopeful that a vaccine would be ready before the end of the year.
"We have got the cavalry coming over the next few months - the vaccine, the mass testing and the improvements in treatments - but we have got to all follow the rules between now and then to keep people safe," he added.
Labour leader Sir Keir Starmer told the BBC he supported the new fines, saying it was important to take action against the few people who were not complying.
He also said he would back a future lockdown in order to reinforce the government's message.
"In the end this is not about party politics, this is about getting the nation through this virus," he added.
The UK government hopes the new fines will be replicated in Wales, Scotland and Northern Ireland - which all have powers to set their own coronavirus rules.
At-a-glance: What are the new rules?
- People in England who are told to self-isolate by NHS Test and Trace face fines of £1,000 - up to £10,000 for the worst offenders - if they fail to do so
- This includes those who test positive and those identified as close contacts of confirmed cases
- It also includes employers who force staff to ignore an order to self-isolate
- NHS Test and Trace will make regular contact with those isolating to check compliance
- The measures apply from 28 September and will be enforced by police and local authorities
- Those in receipt of benefits or on low income and who cannot work from home may receive a £500 one-off payment if self-isolating
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