The UK is on track to meet its target of offering a vaccine to every adult by September, and beating that would be a "bonus", the foreign secretary said.
Dominic Raab reiterated that the plan was to deliver 15 million jabs by mid-February and 17 million more by spring.
Asked about reports that every adult could be vaccinated by the end of June, he told the BBC: "If it can be done more swiftly, that's a bonus."
It comes as 24-hour vaccine centres are due to be piloted in the next ten days.
Mr Raab told the Andrew Marr Show that he was not aware of any delays to supplies from manufacturers Pfizer and AstraZeneca and said he was "confident we have the flexibility" to deliver enough doses.
"It is an enormous challenge. We are meeting it," he said. "But we take nothing for granted."
Saturday's figures showed a further 324,233 people received their first of two vaccine doses across the UK, taking the total above 3.5 million.
More people have now been vaccinated than have had positive tests since the pandemic began, with ten more mass vaccination sites due to open in England on Monday.
Mr Raab said the risk that new variants could prove resistant to vaccines or more deadly meant the UK had to take the "precautionary approach" of requiring all travellers to quarantine on arrival from Monday, closing the travel corridors which previously been exempt.
"We don't want to find in two or three weeks time that our vaccine roll out is imperilled because we haven't taken the precautionary measures on travel corridors," he said.
Checks by Border Force on the passenger locator forms filled out on arrival would be increased, Mr Raab said, as would the follow-up calls by Public Health England intended to ensure people were isolating for up to 10 days.
Asked whether the UK would introduce quarantine hotels to ensure people maintained their isolation, he said all potential measures were under review but there was a challenge in the "workability" of the proposal.
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