Boris Johnson was speaking at the Vaccines Manufacturing Innovation Centre construction site at Harwell.
The Prime Minister has said during a visit to Oxfordshire that a second wave of Coronavirus was inevitable in the UK.
Johnson gave the interview at the Harwell Building site after touring the Jenner Institute in Oxford, which is at the forefront of vaccine research.
The Government’s Scientific Advisory Group for Emergencies (Sage) said the R number – representing the number of people an infected person will pass the virus to – has risen to between 1.1 and 1.4, meaning cases could rise very quickly.
As ministers announced tough new restrictions affecting 13.5 million people, Mr Johnson said they would “keep everything under review”.
He told broadcasters ““Obviously, we’re looking very carefully at the spread of the pandemic as it evolves over the last few days and 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.”
Boris Johnson said a second lockdown was the “last thing anybody wants” but said the current measures would need to be kept “under review”.
“On Monday we brought in the measures that we did, the ‘rule of six’, to really try and restrict what people are doing and to bring in a new buffer – and to make it absolutely clear, the ‘rule of six’: indoors six maximum, six outdoors maximum.
“But the crucial thing is at the same time to observe the basic rules on social distancing – hands, face, space – that is what everybody has got to do if we want to continue to beat this thing.
“But as we look at this particular curve and what is happening now, clearly we are going to keep everything under review. I don’t want to get into a second national lockdown at all, it is the last thing anybody wants.
“I don’t want to go into bigger lockdown measures at all, we want to keep schools open and it is fantastic the schools have gone back in the way they have. We want to keep the economy open as far as we possibly can, we want to keep businesses going
“The only way we can do that is obviously if people follow the guidance.”
The Prime Minister said his administration was considering whether it needed to “go further” than the current national restrictions that were put in place this week.
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.
“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, 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.”