Ministers are “absolutely open” to delaying the date, he said.
While restrictions easing has already been delayed in Scotland over the spread of this variant, first identified in India, a similar move has not yet been made south of the border.
Mr Hancock also said he “would not rule out” measures such as wearing face coverings in public settings and working from home where possible continuing in the long term.
“We said in the road map that June 21 is the date by which we would not take Step 4 before that date and that we would look at the data.
“That is exactly what we are doing, so the road map was set up in order to be able to take these sort of changes into account.”
The Delta variant has made decisions over whether or not to reduce restrictions “more difficult”, Mr Hancock said, as he confirmed latest scientific findings that this variant is 40 per cent more transmissible than the strain first identified in Kent.
The Delta variant has become dominant in Scotland, making up around 75 per cent of new cases.
Nicola Sturgeon has delayed the planned move to level one restrictions in 14 council areas in light of the variant’s spread.
The First Minister warned on Friday that Scotland is at a “critical juncture” in terms of the virus, as she reported 992 new cases, the highest daily total since mid February.
Under the current plan, England is set to remove “all legal limits on social contact” on June 21.
Nightclubs and music venues will be allowed to open, and large events can resume without crowd limits.
Prof Stephen Reicher of St Andrews University warned on Saturday that it would be “foolish” and a “major risk” for England to continue with the June 21 unlocking as planned.