Matt Hancock said "people have got more relaxed over the summer" but "now is the moment when everybody needs to get back" to following the rules.
Sophie Ridge told Sunday that he is "concerned" that too many people are violating self-isolation guidelines.
"I've had discussions this week with the mayor of London, and the teams are meeting today to discuss further what might be needed".
His interview comes as there are reports the United Kingdom could face a second national lockdown.
The Tory MP also advised that people in the United Kingdom should report their neighbours if they see them breaching any of the emergency COVID-19 measures brought in.
Asked if he would, the cabinet minister said: "Yes - and everybody should. It does not allow for play-dates or parties, but it does mean that a consistent childcare relationship that is vital for somebody to get to work is allowed".
Announcing the new rules, Mr Johnson said: "The best way we can fight this virus is by everyone following the rules and self-isolating if they're at risk of passing on coronavirus".
The Chief Medical Officer and the Chief Scientific Adviser will deliver a stark televised address to the nation this morning, with Prof Whitty expected to say that the United Kingdom is facing a "very challenging winter".
Meanwhile, opposition Labour leader Keir Starmer called on the British government to entrust coronavirus testing to local authorities who have a comprehensive understanding of their communities.
When asked if he would report to the police anyone he sees breaking the self-isolation rules, Hancock said he would, because "the self-isolation part is absolutely necessary ... to break the chain of transmission".
A person is tested for COVID-19 at a drive-through testing centre in a vehicle park at Chessington World of Adventures, in Chessington, Greater London, Saturday, Sept. 19, 2020.
It was supported by Mr Hancock, who said Sunday: "We were expecting an increase in demand for people with symptoms".
Pubs who fail to enforce social distancing rules - including making people fill in Track & Trace forms - could be forced to shut, according to the Daily Mail.
Prof Heneghan told Sky News that Covid-19 was operating in a seasonal way similar to other respiratory infections, saying: "What we have to do now is slow down, this is a long winter". "So what we have to do now is go slow".
"And that kind of seems to be the bit that is failing and here we go, it is now down to us to save the day by not seeing anybody and locking down".
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