All restrictions are due to be lifted on 21 June although concerns about the more transmissible B.1.617.2 strain have this week appeared to throw those plans into doubt.
The variant is “dominant” in the Lancashire areas of Bolton and Blackburn and spreading to other areas of the UK.
But ministers have previously said there is no evidence to suggest that it evades vaccines or causes more serious disease in those it infects.
The health secretary, Matt Hancock, earlier this week said that most of those in hospital in Bolton had not taken up the offer of a vaccine as he refused to rule out local lockdowns.
Addressing reporters at a vaccination centre in London on Tuesday, the prime minister said he had seen “no conclusive evidence” to signal that the planned easing on 21 June could not go ahead.
He warned that “we’ve got to be cautious” but stressed that the “wall of defences” built up by the vaccination programme put England in a much better position than in previous lockdowns.
“I don’t see anything conclusive at the moment to say that we need to deviate from the road map”, Mr Johnson said. “We’ll know a lot more in a few days’ time.”
Mr Johnson said data from hotspots including Bolton, Blackburn, Bedford and Sefton were being examined to find out more about the impact of the variant.
Asked whether local lockdowns could be used, Mr Johnson added: “We’ve just got to be cautious about the way we approach it and we will be letting people know as much as we can, as soon as we can.”
Mr Johnson’s comments came as Britons jetted off for holidays in Europe and elsewhere, many of them to “amber list” countries following the relaxation of travel rules.
Under the new traffic light system, countries are rated as red, amber or green, depending on the risk they are believed to pose of importing new cases and variants of coronavirus to the UK.
Downing Street has faced increasing pressure in recent days over what critics have described as “leaky borders”, after reports that thousands of people flew out of Britain on up to 150 flights to amber list sunspots such as Greece, France, Spain and Italy on Monday.
Addressing those concerns on Tuesday, and seeking to clarify confusion after a senior minister said people could visit amber list countries to see friends and family, Mr Johnson said amber list countries are not “somewhere where you should be going on holiday”.
He added: “And if people do go to an amber list country – they absolutely have to for some pressing family or urgent business reason – if they have to go to an amber list country, then please bear in mind that you will have to self-isolate, you’ll have to take tests and do your passenger locator form and all the rest of it, but you also have to self-isolate for 10 days when you get back.”
He added: “And that period of self-isolation, that period of quarantine, will be enforced with fines of up to £10,000 so I think it’s important for you to understand what an amber list country is.”
Environment secretary George Eustice had earlier in the day stoked confusion about travel rules, saying amber list holidays were allowed.
His comments contradicted those of Mr Hancock, who on Sunday said: “The red and amber list countries are places that you shouldn’t go to unless you have an absolutely compelling reason.”
No 10 spokesman said: “The position remains that people should not travel to amber list countries and that’s to protect public health.
“There are some limited reasons why it might be acceptable to travel – for work purposes, protecting essential services or compassionate reasons such as a funeral or care of a family member but otherwise people should not be travelling to these countries.”
Labour said the government’s Covid travel policy and its handling of borders had descended into “dangerous chaos”.
Nick Thomas-Symonds, the shadow home secretary, suggested said that a “lack of strategy” had led to conflicting advice among ministers.
He added: “Labour has been clear that there should be a pause on international travel, to guard against further importing of dangerous strains, setting back hopes for ending restrictions.”