Matt Hancock has warned it is "too early" to say if lockdown will end after December 2.
The Health Secretary said there is still a "very high" number of coronavirus cases.
But he insisted the Government wants to end the national lockdown in England, which has forced all pubs, bars, restaurants and non-essential shops to shut.
The lockdown is currently due to end on December 2, with plans for a return to the tier system of alert levels.
Asked if the lockdown would simply be "rebadged" after the deadline, Mr Hancock said: "You tempt me, but it is too early to say I'm afraid.
"We've seen in the last week that there is still a very high number of cases but we do absolutely want to come out of this national lockdown."
He told BBC Radio 4's Today programme: "That is our goal, everybody has a part to play in making that happen of course, following the social distancing rules and isolating when you need to, which is the critical thing."
One of the main goals now was to use the mass rapid testing roll-out to find those who are asymptomatic with the virus, he added.
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He also pledged that all care homes will be able to allow visits before Christmas.
Rapid tests for visitors, which are currently being trialled in Devon and Cornwall, should allow rollout across England within weeks, the minister said.
Meanwhile, Boris Johnson is self-isolating after coming into contact with infected MP Lee Anderson last week.
He was told to self-isolate by NHS Test and Trace on Sunday.
But the Prime Minister uploaded a video to Twitter on Monday claiming he was "bursting with antibodies" and "fit as a butcher's dog".
He said: "We've got to interrupt the spread of the disease and one of the ways we can do that now is by self-isolating for 14 days when contacted by Test and Trace."
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