The delay to scrapping lockdown restrictions could be halved to two weeks following a "'genuine review of the data."
A government spokesman reportedly revealed restrictions could end on July 5 if data is "much better than expected."
Earlier this week, Prime Minister Boris Johnson announced lockdown would continue until July 19 - four weeks later than previously expected - after the Delta variant began to spread across the UK.
But there are hopes the deadline could now be brought forward by a fortnight, with a two-week review of the situation due on June 28, the Mirror reports.
"If we get to the end of next week and the data is moving decisively in the right direction, no-one is going to criticise us for changing our minds and opening up a bit early," a Cabinet source is reported to have said.
However, the government spokesperson stressed an early exit from lockdown remained "unlikely".
The delay came after new modelling by the Scientific Pandemic Influenza Group on Modelling (SPI-M) - a SAGE subgroup - revealed how risky scrapping all social distancing could have been.
Among the experts' worst case scenarios was that hospitalisations would reach around the peak of the first wave, when there were more than 3,000 new UK patients per day, compared to fewer than 200 a day now.
While ministers believe re-opening on July 19 is still the most likely scenario, the two-week review will be a "genuine review of the data," according to reports.
"The decision to delay reopening was so finely balanced – probably the most difficult decision of the whole pandemic – that the PM wanted a review point built in so that if things did change we could move sooner," a source told the Mail.
"No-one wants these restrictions in place for a day longer than necessary."
The PM's official spokesman said earlier that throwing open the economy as planned could have meant a “surge in hospitalisations which would put unsustainable pressure on the NHS”.
In the worst scenario estimates showed a further 203,824 deaths by next June, with more modest estimates suggesting more than 50,000 would die.
But reports earlier this week claimed that the models were based on 'out of date' estimates of vaccine effectiveness, which was said to have assumed fewer people were protected by jabs.
A Cabinet source was reported to have added: "If we get to the end of next week and the data is moving decisively in the right direction, no-one is going to criticise us for changing our minds and opening up a bit early."
It comes after recent figures revealed the latest Covid hotspots with stark numbers showing nearly 90% of England's local authorities have seen a weekly rise in infections.
The grim tally, for the seven days to June 12, showed that of the 315 local areas in England 279, representing 89%, saw a rise in case rates.
Just 35, or 11%, of local authorities saw a fall. One remains unchanged.
Blackburn with Darwen in Lancashire continues to have the highest rate, with 863 new cases in the seven days to June 12 - the equivalent of 576.5 cases per 100,000 people.