Communities Secretary Robert Jenrick said the government was "very sympathetic" to the idea of inviting children aged 12 to 17 to have a jab.
It comes as every adult in the UK has now been offered a vaccine, with the prime minister hailing the "extraordinary achievement".
England and Scotland are set to ease restrictions from Monday.
It comes as more than 50,000 daily coronavirus cases were recorded on both Friday and Saturday - levels last seen in January.
So far, around 88% of adults have had a first dose and around 68% have had both, according to the latest figures.
The Joint Committee on Vaccination and Immunisation (JCVI) advises government ministers on which people should be offered a vaccine.
Mr Jenrick told the BBC's Andrew Marr Show: "We are still awaiting the final advice from the JCVI about extending the vaccination programme to younger people.
"It seems like a sensible thing to do. The evidence we have received so far is compelling and ministers are going to make a decision armed with the advice in the coming days."
If it does get the go-ahead, those just short of their 18th birthday are likely to be invited to have a vaccine first, along with those who have health vulnerabilities and children who live in households with others who are more vulnerable.
His comments come after a report in the Sunday Telegraph said that the JCVI is believed to have advised ministers against a mass vaccination programme for all children until further evidence of the risks is available.
According to the Telegraph, guidance this week is expected to say the vaccine should be offered only to vulnerable 12 to 15-year-olds and to those who are within three months of turning 18.
More than 46 million people have been given a jab since the UK's Covid vaccine rollout began in December 2020.
Mr Johnson promised in February that all adults would be offered a first dose by the end of July, and the Health Secretary Sajid Javid said last month that the government wanted two-thirds of UK adults to have had two by 19 July.
The prime minister urged anyone who had not booked their vaccine to do so and thanked those who had taken up their offer.
"You are the reason we are able to cautiously ease restrictions next week, and return closer towards normal life," he said. "Now let's finish the job."
But in Scotland, almost a third of younger adults remain unvaccinated - despite the fact that by the end of Sunday, all over-18s will have been offered an appointment to have their first dose.
Health Secretary Humza Yousaf said some had re-arranged appointments and would receive the jab shortly.
The UK recorded 54,674 Covid cases on Saturday - following 51,870 new cases on Friday - as well as 41 deaths within 28 days of a positive test.
The last time cases exceeded 50,000 was 15 January.
Health Secretary Sajid Javid, who has been doubled jabbed, announced on Saturday he had tested positive for coronavirus and was isolating. It is understood Mr Javid met Boris Johnson at Downing Street on Friday - the prime minister is now self-isolating at Chequers.
Chancellor Rishi Sunak was also contacted by NHS Test and Trace and is isolating.
Almost all legal restrictions on social contact are to be lifted in England on Monday.
New travel rules will also come into effect, allowing people who are double-jabbed to return to the UK from amber list countries without quarantining - except for those returning to England, Wales and Scotland from France, because of concerns over the prevalence of the Beta variant there.
The prime minister has described England's approach to easing lockdown as "cautious but irreversible".
But Jeremy Hunt, chairman of the Commons health select committee, said on Saturday the government was likely to have to reintroduce some controls in the autumn, as rising case numbers and hospital admissions meant the NHS was facing a "very serious" situation.
England's deputy chief medical officer, Prof Jonathan Van-Tam, warned of a "bumpy winter" ahead. He urged people to approach the easing of restrictions in "a cautious, steady, gradual way".
Social distancing rules will end in England on Monday, although government guidelines advise face coverings should still be worn in enclosed spaces such as in shops and on public transport.
Scotland will move to level zero of Covid restrictions on Monday, meaning pubs and restaurants can open until midnight. However, limits on outdoor meetings will remain, the return of workers to offices will be delayed and face coverings will still be mandatory.
Most Covid rules in Wales are set to be scrapped from 7 August., but face coverings will still be required in most public places and on public transport.
In Northern Ireland, restrictions will be eased further on 26 July, if approved at a review on 22 July.