The Treasury is said to be prioritising the end of the “one metre plus” distancing rule, as well as the “rule of six” indoors to help boost hospitality and retail businesses.
On Friday, the UK recorded 4,000 coronavirus cases for the first time since the end of March, while the R number moved above one for the first time since January.
The R value is the average number of people an infected individual will pass the virus on to. If it rises above one it means the pandemic is growing.
Business secretary Kwasi Kwarteng has already said the government “can't guarantee” the final stage of lifting lockdown measures will go ahead as planned.
If the Indian variant - known to be more transmissible than previous dominant strains in the UK - causes a spike in hospital admissions the June 21 date may have to be pushed back.
However, Mr Kwarteng did say that at the moment there was “nothing in the data” to suggest it will be postponed.
In a bid to help reduce the spread of the virus masks could still be required on public transport after June 21.
Guidance stating people must work from home if they can may also stay in place.
Boris Johnson is expected to make a decision about which restrictions can be lifted in the next fortnight.
Meanwhile, an expert warned on Saturday that “confusion” over the government’s handling of Covid restrictions is undermining efforts to control the virus.
Professor Stephen Reicher, a psychologist on the Sage sub-committee advising ministers on behavioural science, said the government is in a “pickle” because it appears to have abandoned the “data not dates” principle.
Speaking to BBC Radio 4’s Today programme, Prof Reicher said: “I think we are in a pickle of the government’s own making at the moment.
“I think the reason for that is it has departed from its own mantra of ‘data not dates’.
“Very quickly ‘data not dates’ became ‘dates not data’.
“People were promised that things would happen on particular dates and they invested so much in them, and the government invested so much political capital in them, that it became very difficult to do anything else if the data suggested it was unwise.”
Current data suggests that, although hospital admissions are rising in some parts of the country affected by the Indian variant, overall admissions remain broadly flat.