He said: "There are some baseline measures that can remain in place without drastically affecting the economy.
"A lot of people can relatively easily work from home without it affecting their productivity and having a huge economic impact, and that would substantially reduce the amount of transmission."
Prof Hayward warned cases of the new strain are doubling every week and could reach a quarter of a million by mid-July.
He added that some restrictions may have to remain in place until a greater proportion of the population is fully vaccinated against the virus.
More than 70 per cent of Brits have had one dose and just over a third have received two.
He said: "It's still going to be a few weeks yet until we've got all of the highly clinically vulnerable double vaccinated.
"That will about coincide with when we plan to open more fully up.
"When we do open up more fully then instead of doubling every week it's likely to double more frequently than that.
"There's a good argument for caution until such a time as we've got a much higher proportion of the population double vaccinated."
It comes as Business Secretary Kwasi Kwarteng said No 10 "can't guarantee" the June 21 reopening will go ahead as planned.
"We’re very attentive to the data and we don’t want to proceed with the roadmap without considering all the information that we have,” he said.
“The numbers of cases is a matter of concern but what does give me some confidence is the fact the vaccine seems to be working.
"The effectiveness of the vaccine is what can give us some confidence we can reopen on June 21.
“The other thing that's very helpful and encouraging is the number of hospitalisations and fatalities is considerably lower than it was just two months ago.
"We want to reopen on June 21 and we’re trying to get to that outcome but if the data suggests it would be unsafe to do that we would follow that.”
He added: "As of today I can assure people there's nothing in the data that suggests to me we should move the date. The caveat is the data can change.
"If scientific evidence points to an increased hospitalisation rate, increased degree of risk, then we have flexibility to move that date.
"I don't think we will move the date but I cannot guarantee that."