Video report by ITV News Political Correspondent Libby Wiener
His comments come as a new national lockdown is being considered as a "last line of defence".
The prime minister also warned the government was considering whether it needed to "go further" than the current 'Rule of Six' national restrictions which were put in place to limit the spread of Covid-19.
He added that a second national lockdown was the "last thing anybody wants" but said the current measures would need to be kept "under review".
Watch ITV News Health Editor Emily Morgan's live analysis on the latest developments:
Speaking during a visit to the Vaccines Manufacturing Innovation Centre construction site near Oxford, Mr Johnson told broadcasters: "There’s no question, as I’ve said for several weeks now, that we could expect (and) are now seeing a second wave coming in."
"We are seeing it in France, in Spain, across Europe - it has been absolutely, I’m afraid, inevitable we were going to see it in this country," the prime minister added.
He added that coronavirus spreading “among young people” through a lack of social distancing has caused the recent surge of cases.
When asked if the Government had eased the national lockdown too early, said it was “difficult” to maintain the “discipline” of the measures.
“If you look at what’s happened over the last few months, I think the British people have done an amazing job,” Mr Johnson said.
“They got that peak under control, they brought it right down, they brought the number of infections right down by discipline and everybody adjusting our behaviours and the way we go about our lives – hands, face, space.
“And I think probably, truth to tell, what’s happened here and what alas has happened in so many other countries is that people find it difficult to keep this up.
“It is very difficult to maintain that kind of discipline for a long time and what you’ve seen is the disease starting to spread again among young people and that’s where it really started to kick off in France and Spain in those age groups and we are now seeing that here in this country.”
The prime minister's visit comes as the government earlier published the latest coronavirus figures including how the UK coronavirus death toll has risen by 27 and 4,322 Covid-19 cases were recorded in the past 24 hours.
The latest numbers suggest there is a risk that the overall coronavirus epidemic in the UK is increasing, not decreasing.
The reproduction rate of coronavirus (R number) has increased to above 1.0 in virtually all parts of England - including London, the Midlands, North East and Yorkshire, the North West and South East regions.
ITV News Political Correspondent Libby Wiener explains the latest developments with the lockdown restrictions across England
Public Health England (PHE) warned that data published on Friday could be a sign of "far worse things to come", as the Office for National Statistics (ONS) said cases are thought to have almost doubled in a week to 6,000-a-day in England.
Kevin McConway, emeritus professor of applied statistics at the Open University, described the latest R figure as "undoubtedly concerning".
He said: "Even at growth rates within the estimated range, the number of new cases could grow to high levels quickly if the interventions are not sufficiently effective."
Overall, around 13.5 million people are currently living under restrictions or will be from Tuesday.
Health Secretary Matt Hancock stressed the focus remained on local interventions, as he announced new restrictions for large parts of England’s North West, West Yorkshire and the Midlands.
He pointed to a worrying rise in cases, with Merseyside, Warrington, Halton and Lancashire – excluding Blackpool and Greater Manchester – escalated to “areas of intervention”.
When asked about the possibility of a two-week October half-term in order to bring in a short lockdown, Mr Johnson told reporters: "What I can certainly say about parents and schools is we want to keep the schools open, that is going to happen."
Mr Johnson said: "We want to try and keep all parts of the economy open as far as we possibly can – I don’t think anybody wants to go into a second lockdown but clearly when you look at what is happening."
He added: "You have got to wonder whether we need to go further than the 'rule of six' that we have brought in on Monday, so we will be looking at the local lockdowns we have got in large parts of the country now, looking at what we can do to intensify things that help bring the rate of infection down there, but also looking at other measures as well."
"What I will say is, as we go forward, we will be explaining in great detail to people what the scientific background is, what the epidemiology is saying and really how we propose to do it."
Mr Johnson also reiterated that people should keep up with wearing a face covering and washing their hands, while only those with Covid-like symptoms should go for a test.