Sir Jeremy Farrar, director of the Wellcome Trust, admitted he is "worried" about a possible spike in infections ahead of the reopening of pubs, restaurants and hairdressers next month.
He warned that there could be a "very nasty rebound" of coronavirus in the winter if the UK does not use the next few months "sensibly".
Speaking to the BBC's Andrew Marr Show on Sunday, he said: "In truth, the restrictions started to be lifted towards the end of May, the beginning of June, around that bank holiday.
"I would predict, I would guess, that we will start to see a few increases in cases towards the end of June or the first week of July.
He added: "We're on a knife edge, it's very precarious the situation, particularly in England at the moment, and I would anticipate we would see an increase in new cases over the coming weeks."
Sir Jeremy, who is also a member of the Government's Scientific Advisory Group for Emergencies (Sage), argued that the next three months were "absolutely critical" in the fight against coronavirus in the UK.
He said: "Come the winter, come the reopening of schools, which is absolutely critical, we can anticipate to see rebounds and second waves.
"The question is do you start from a very low base, like in Scotland, a few dozens cases, or maybe a few hundred cases in England, and then you're in a good position if there are local outbreaks that you can respond locally and you can prevent the national catastrophe that happened in March and April.
"That's the key, using June, July and August really cleverly, making sure we have everything in place and learning the lessons from February and March."
He said that doctors had got better at treating patients with Covid-19, but it still represented a "very, very nasty infection".
Sir Jeremy added: "You've got to move faster than the epidemic, you've got to move ahead of the pandemic.
"Once you get behind it, you have exponential growth, you lose track of it and that's what happened across Europe, particularly in the UK, in March and April of this year.
"We got behind it, we were too slow and, as a result, the epidemic took off and we weren't able to control it."
Sir Jeremy argued the Government's test, trace and isolate system needed to be "fully functional" by September.
Sir Mark Walport, the former government chief scientific adviser, said the UK needs to maintain "constant vigilance" as it eases out of lockdown.
Speaking to Sky's Sophy Ridge on Sunday, he said the Government faced a "fine balancing act" between managing the virus outbreak and the health harms caused by a damaged economy.
Asked if the virus could come back in winter when the NHS is under more pressure, Sir Mark said: "That is obviously a significant risk."
He said that the virus probably lasts longer in the air and on surfaces in cold and wet environments, suggesting that the winter could be a "risky" time.
Sir Mark said people needed to be "sensible and responsible" and to try and reduce social contact as much as possible.
He suggested outbreaks were occurring in "clusters", including in certain work environments such as food processing factories, making local control important.
"We need to do everything we possibly can to avoid a widespread second wave," he said.
Sir Jeremy urged people to still be "really cautious, particularly around any events indoors", as lockdown measures are eased.
He said that easing lockdown at the end of May was "too early" due to the number of new coronavirus cases per day.
But he told the BBC's Andrew Marr Show that lower numbers now made it "reasonable" to open up the economy.