Chancellor Rishi Sunak was willing to accept a short delay to Step 4 of the road map to ending the lockdown amid a rise in cases, it is understood.
A Whitehall source pointed towards the Treasury having gone “long” on emergency coronavirus support packages in the Budget to cover the possibility of a delay to the plans.
It comes as Tory lockdown-sceptic Sir Charles Walker warned Prime Minister Boris Johnson against delaying the earmarked end to legal coronavirus restrictions.
The vice chairman of the 1922 Committee of Conservative MPs told BBC Newsnight: “There will be a huge wave of disappointment across the country if we don’t open up on June 21.
“The delay could be two weeks, a month, but I think the real issue here is if we can’t open up the economy at the height of summer then I think we are facing the very real prospect of more forced lockdowns in the autumn – I just don’t see how we can avoid that.
“The goalposts – as we’ve always said, it’s a well-worn cliche – are moving.”
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The government said it was a "finely balanced decision" and asked the public to be "patient for a few more days" before the decision is made.
Secretary of State for Housing, Communities and Local Government, Robert Jenrick, told ITV News: "If we can reopen further on June 21, of course we will do so."
When asked if the Chancellor's views about the lockdown meant that the easing would have to be delayed, Mr Jenrick said: "We haven't made a decision yet.
"We are looking at the data, data is still coming in.
"We're seeing cases start to rise but we're also seeing that the link between those cases and hospitalisations...is at least disrupted, hopefully broken."
He added the government would make a final judgement "next week."
Health Secretary Matt Hancock said on Tuesday there was a “challenging decision” to be made over the further lifting of Covid restrictions on June 21.
Mr Hancock also announced that a “strengthened package of support” will be provided for Greater Manchester and Lancashire, similar to that seen in Bolton, where case numbers remain high.
It means 5.7 million people are now under what mayor of Greater Manchester Andy Burnham stressed was “not a lockdown” but “advice”.
As part of Government recommendations, people in areas affected by a rise in the variant first identified in India, are being urged to meet outside rather than inside where possible, keep up social distancing and minimise travel in and out of affected regions.
The help on offer includes military support with testing and supervised in-school testing, and greater communication with disadvantaged groups will be available.
Local directors of public health will also be able to reintroduce face coverings in communal areas in schools if they want to.
But Mr Burnham called on the Government to go further, and urged ministers to release vaccine supplies earlier than planned.
Speaking at a press conference, he said: “Obviously what we’re seeing here is a localised approach to messaging, more localised support on testing and on tracing and isolation. We are also saying that also should apply to vaccination.
“We are not asking for any more vaccine here than our fair share, what we are asking for is the bringing forward of Greater Manchester’s supplies, so that we can run a surge vaccination programme over the next three weeks.”
Earlier, former health secretary Jeremy Hunt said any delay to England’s road map for easing lockdown would only be for a couple of weeks, owing to the success of the vaccination programme.
Mr Hunt, who chairs the Health and Social Care Committee, said he was “feeling quite optimistic that we are going to see the restrictions lifted” as “being double-jabbed” works against the Indian variant.
He told Times Radio: “If freedom day ends up being put back a couple of weeks so we can get more people double-jabbed, I think it will only be a temporary setback.”
In a further boost to the vaccine programme, NHS England said nearly 500,000 jabs were booked on Tuesday in a “Glastonbury-style” rush after the vaccine rollout was expanded to 25 to 29-year-olds.
NHS England said the National Booking Service had seen 493,000 appointments reserved by midday on Tuesday, just five hours after eligibility was widened to the over-25s.
It is more than double the number booked the previous day and equates to about 100,000 bookings an hour.
NHS England chief executive Sir Simon Stevens said: “This encouraging Glastonbury-style rush for appointments has already now seen hundreds of thousands of people between 25 and 29 book in for their NHS Covid jabs, as more vaccine supplies continue to come on line.
“Pleasingly this suggests strong enthusiasm for vaccination amongst people in their 20s, following hard on the heels of the millions of others who’ve already taken up our offer.”