"The virus probably will mutate. Different vaccines will be needed at that point," he said.
Johnson also refused to confirm that schools would reopen before the summer holidays but insisted that it was a top priority.
England is set to remain in lockdown until at least March, and some measures could stay in place for even longer, as hospitals struggle to cope with the highly contagious new strain of the coronavirus.
Announcing the lockdown on Monday, Prime Minister Boris Johnson said the restrictions were expected to last until the middle of next month, when the UK's National Health Service (NHS) hoped to have vaccinated priority groups, including older care home residents and staff, over 70s, frontline health workers and the clinically extremely vulnerable.
But Cabinet Minister Michael Gove told Sky News on Tuesday that "we can't predict with certainty that we will be able to lift restrictions in the week commencing February 15 to the 22".
"What we will be doing is everything that we can to make sure that as many people as possible are vaccinated, so that we can begin to progressively lift restrictions," he said.
"I think it is right to say that as we enter March we should be able to lift some of these restrictions, but not necessarily all."
He called that "a significant, a stretching target" but one that could be achieved.
Yesterday the UK recorded 60,916 new cases, its highest daily total.
It also recorded 830 deaths, bringing the death toll from the past seven days to 4738 - a 45 per cent rise in deaths compared to the previous week.
- Additional reporting, news.com.au