It comes as concerns grow over the spread of the Indian - or B.1.617.2. variant - in Covid "hotspots", particularly in London and the northwest of England.
Ministers are said to be considering "surge vaccinations" of all younger adults in areas with the highest numbers of cases.
Mr Johnson's spokesman did not rule out that possibility, saying earlier he would "not get ahead" of his boss.
Vaccines minister Nadhim Zahawi urged people in the areas where the strain is spreading to follow local health advice, get tested and isolate if they test positive.
London and northwest of England have seen the biggest rise in cases, with Public Health England data showing the Indian variant has been responsible for four deaths as of 12 May.
Asked if the 21 June road map easing - when all legal limits on social contact are due to be lifted - could be put on ice, Mr Zahawi told LBC's Nick Ferrari that this Monday's reopening of indoor meet-ups "is still on".
When pressed on whether the plans for June 21 could be paused, he said: "The way we don't have to do that is by everybody doing their bit, by taking the two tests a week, doing your PCR test in those areas, and to isolate, isolate, isolate.
"We have got to break the cycle of infection, because one of those big tests was infection rates have to be suppressed, and the other big test is variants.
"If those cause a problem, then the tests will fail. The four tests have to be met for 21 June.”
Mr Zahawi said the seven-day rolling average figures for infection show a 12.4 per cent rise, but hospital admissions are down by 7.9 per cent.
"That is good news because it tells you that the vaccines are clearly working in terms of hospitalisation and severe infections... but the infection rate is what is concerning, which is why we have to surge test and then isolate," he said.
He also told the programme there is no evidence to show the India variant escapes vaccines or has a more severe impact on the people it infects.
Additional reporting by Press Association