Former government adviser Prof Neil Ferguson said new measures were needed "sooner rather than later".
The scientist, whose advice was crucial to the decision to go into lockdown in March, said: "If we leave it another two to four weeks, we will be back at (infection) levels we were seeing more like mid-March.
"That's clearly going to cause deaths because people will be hospitalised," he told the BBC.
Johnson only has direct power over restrictions in England. The devolved governments in Scotland, Wales and Northern Ireland can set their own rules.
Widespread growth of the virus has been recorded across the country, with cases doubling every seven to eight days.
There were a further 4,422 confirmed UK cases on Saturday and 27 deaths. Friday was the first time the daily total of positive tests had exceeded 4,000 since May 8.
A further 350 cases were reported on Saturday in Scotland, the highest daily increase since May, 212 new cases in Wales, and 222 in Northern Ireland.
In Bolton, one returning holidaymaker, who did not self-isolate and instead went on a pub crawl, is being partly blamed for the town's spike in cases.
Meanwhile, governments across Europe have announced new restrictions to tackle the surge in infections.
Speaking on Friday, Johnson said he did not "want to go into bigger lockdown measures" but that tighter social distancing rules might be necessary.
He added: "Clearly when you look at what is happening, you have got to wonder whether we need to go further than the rule of six that we brought in on Monday."
The government is understood to be considering a short period of tighter rules across England - "a circuit break" - which could involve closing hospitality venues.
However, schools and workplaces would stay open.
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