Video report by ITV News Political Correspondent Shehab Khan
The prime minister said: "If you look at the numbers there's no question we will have to take tougher measures and we will be announcing those in due course."
He said there are "tough, tough" weeks to come in the battle with Covid-19, with a new South African variant of the virus adding worries about fast-spreading variations of coronavirus.
ITV News Political Editor Robert Peston has more on what the tougher measures are likely to be:
Speaking during a visit to Chase Farm Hospital in north London to meet some of the first people to receive the Oxford vaccine, Mr Johnson said the government "will do everything we can to keep the virus under control".
"People should be in no doubt that the government will do everything that's necessary," he said, "but I must stress at this critical moment it is so vital that people keep disciplined."
It comes as Scotland's First Minister Nicola Sturgeon announced a nationwide lockdown in the country to curb the rise in Covid-19 cases.
The prime minister's comments follow the release of some worrying coronavirus statistics on Sunday, which showed more than 50,000 people had tested positive for Covid-19 for the sixth day in a row.
The figures come amid speculation about a new set of stricter coronavirus restrictions, being labelled as 'tier 5' on social media, which could see a limit put on how long people can leave their homes.
Downing Street said the government is waiting to see the impact of the latest coronavirus restrictions in England before deciding on which further measures would be needed.
The PM's official spokesman said: "We have been waiting to see the impact of the Tier 4 measures. It is a bit unclear still at the moment. "
Boris Johnson on the need for tougher measures:
He added: "We have always said that we would take the measures needed to reduce the spread of the virus and we will continue to do."
It is hoped coronavirus vaccines will provide a route out of strict restrictions, with the Oxford University and AstraZeneca Covid-19 jab following Pfizer/BioNTech in a UK roll-out.
The first doses of the vaccine were administered on Monday morning in what has been described as a "pivotal moment" in the UK’s fight against coronavirus by Health Secretary Matt Hancock.
Dialysis patient Brian Pinker was the first person in the country to be given the jab outside of clinical trials, at Oxford University Hospital, NHS England said.
Mr Johnson acknowledged issues will rolling out coronavirus vaccines, but said there's a "massive ramp up operation now going on".
The PM said the limiting factor in expanding the UK's vaccine rollout was not supply or staff but waiting for each batch to be approved.
"We have the capacity," he said, "the issue is to do with supply of the vaccine".
He added: "The rate limiting factor is now not supply of vaccines although we want that to go faster, it's getting them properly tested and getting them to the NHS.
"It's not the ability to distribute the vaccine, it's not the shortage of staff.
"It's getting it properly tested. That will ramp up in the weeks ahead."
It is not yet clear whether newly-approved coronavirus vaccines will have an effect on new variants of the virus.
Health Secretary Hancock said he is "very worried" about the South African variant of Covid-19 and that the country should keep an "eagle eye" on it.
Matt Hancock on the South African variant of Covid-19:
The South African variant is considered to be more transmissible than the new, fast-spreading strain of the virus which was recently identified in the UK.
ITV News Political Editor Robert Peston said that according to one of the government's scientific advisers, the reason for Mr Hancock's "worry" about the South African Covid-19 variant is that they are not as confident the vaccines will be as effective against it as they are for the UK's variant.
Scientists at Porton Down are currently looking into the impact of vaccines on the new strains.
Labour leader Sir Keir Starmer on Sunday urged the PM to enforce a national lockdown within 24 hours, warning "the virus is clearly out of control".He said national restrictions have to be "the first step towards controlling the virus".