Lord Sedwill, who left his job last month, told the BBC there is “a genuine question” about whether the UK could have been “better prepared” for the pandemic.
He also said Dominic Cummings’ journey to County Durham during lockdown had been a mistake.
And he admitted to feeling “troubled” by attacks on the civil service.
Lord Sedwill stepped down as the UK’s top civil servant following reports of tensions between him and senior members of Boris Johnson’s team.
Reflecting on the government’s handling of coronavirus, he said: “Although we had exercised and prepared for pandemic threats, we didn’t have in place the exact measures, and we hadn’t rehearsed the exact measures” for the challenge Covid-19 presented.
“I think there is a genuine question about whether we could have been better prepared in the first place and that is obviously a very legitimate challenge.”
He said any future inquiry would have to look at whether decisions were taken at the right time, if the lockdown was imposed fast enough and what capabilities the state had to deploy to tackle the virus.
Lord Sedwill, who contracted the virus himself, said he was “really proud of a great deal that we did” including setting up the Nightingale Hospitals.
In May, the prime minister’s chief adviser, Dominic Cummings, said he had acted “reasonably” when he drove to County Durham after his wife developed Covid-19 symptoms.
Asked about the incident Lord Sedwill said that “it was clearly a difficult moment for the government”.
“It was a mistake – whether everyone should quit every time they make a mistake, I don’t think is right.
“But it clearly undermined the government’s coherent narrative about people following the rules.”