Michael Gove says it is his “fervent hope” that England’s new lockdown will end on 2 December, but ministers will be “guided by the facts”.
“We do need to get the R rate below 1,” the Cabinet Office minister told the BBC’s Andrew Marr Show.
The strict measures are due to come into force from Thursday.
Pubs, restaurants, gyms, non-essential shops and places of worship will close, but schools, colleges and universities can stay open.
The prime minister is expected to deliver a statement in the Commons on Monday ahead of a vote on the latest restrictions on Wednesday. Labour has indicated it will back the lockdown.
Boris Johnson said he expects the lockdown to last until 2 December, after which England’s regional tiered system would be reintroduced.
But Mr Gove told the BBC decisions would “obviously be guided by the facts”.
He said ministers believe “on the basis of the evidence that we have that we will be able to [lift restrictions]” by 2 December, but stressed that “we do need to get the R rate [the number of people that one infected person will infect] below 1”.
Mr Gove added that “we do need to be vigilant” and said “the reason why you need to take these steps is because of the danger of the NHS being overwhelmed”.
Earlier, he told Sky News the lockdown could be extended beyond the December deadline.
Labour leader Sir Keir Starmer told the BBC there would “be no effective exit on 2 December unless the government uses this time to fix test, trace and isolate”.
He has called for the lockdown to stay in place until the infection rate is below 1.
The UK recorded another 21,915 confirmed coronavirus cases on Saturday, bringing the total since the pandemic began to 1,011,660.
Another 326 people were reported to have died within 28 days of a positive test.
The UK is the ninth country to reach the milestone of a million confirmed cases – after the US, India, Brazil, Russia, France, Spain, Argentina and Colombia.
Almost 11,000 people are now in hospital with Covid-19, including 978 on ventilators.
A presentation by the government’s pandemic modelling group SPI-M shown to the PM and seen by the BBC included one forecast which suggest
