More than half a million people signed up in just 24 hours to help vulnerable people self-isolating for the next 12 weeks.
Despite the threshold being met twice over - with 560,000 people volunteering by Thursday morning - the Prime Minister's official spokesman said the scheme would stay open.
The spokesman told a Westminster briefing: "I think the most recent number I've seen was around 560,000 people who have volunteered so far which is amazing - but we want people to continue to volunteer and if we can reach a number of 750,000 then that would be fantastic."
Health minister Edward Argar told ITV News the drive showed the country "at its best".
He added: "We need to come together to beat this disease and when this country comes together as it is doing, this country is unbeatable."
Volunteers are being sought to help the 1.5 million people isolating for 12 weeks in an attempt to slow the spread of Covid-19.
On Thursday, Health Secretary Matt Hancock tweeted: "NEWS: Fantastic that 560,000 people have now responded to our call to volunteer to support our NHS to defeat #Coronavirus."
"When we launched the appeal last night, we hoped to get 250,000 volunteers over a few days."
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Professor Stephen Powis, NHS national medical director, said: "We are truly amazed by the number of people who want to come and help us in the war against coronavirus.
"I would like to thank every last one of you who are offering your time - you will without a doubt help us to save lives.
"Our NHS needs you – whether it's by volunteering or simply staying at home – you are helping your families, communities and protecting our NHS."
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GPs, doctors, pharmacists, nurses, midwives, NHS 111 advisers and social care staff will all be able to request help for at-risk patients through a call centre run by the Royal Voluntary Service.
The service will then match people who need help with volunteers who live nearby.
NHS chief executive Sir Simon Stevens praised the "overwhelming response" to the call for volunteers to help the vulnerable and thanked those who will devote their time to the health service.
"Times like this show just how generous the British people are and how much they value our health service - we are blown away by this response and the kindness of our country," he said.
"I can’t thank those enough who have pledged to devote their time to helping others at what is a challenging and uncertain time for you and your families.
"The NHS is always there for you - now is your time to be there for us too."
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