Boris Johnson, speaking at the daily press conference inside Number 10, said he wanted to offer a "special thank you to everyone who has now volunteered to help the NHS".
"When we launched the appeal last night, we hoped to get 250,000 volunteers over a few days,” he said.
"But I can tell you that in just 24 hours, 405,000 people have responded to the call."
The figure represents more than 16,000 people signing up every hour since the scheme was announced.
Mr Johnson said: "They will be absolutely crucial in the fight against this virus. That is already, in one day, as many volunteers as the population of Coventry.
"And so to all of you, and to all the former NHS staff who are coming back now into the service, I say thank you on behalf of the entire country."
Health Secretary Matt Hancock yesterday called for 250,000 people to sign up as volunteers.
But earlier in the day Professor Stephen Powis, NHS England's medical director, said: "Overnight 170,000 people have signed up - that's three a minute to help the NHS.
"It's an absolutely astonishing response."