The volunteers will be deployed to assist the 1.5 million people isolating for 12 weeks in an attempt to slow the spread of Covid-19.
NHS England medical director, Stephen Powis, said: "Overnight 170,000 people have signed up - that's three a minute to help the NHS."
The senior doctor described the sign up rate as "an absolutely astonishing response."
He told BBC Breakfast: "This is a health emergency, we can all play a role in ensuring we get on top of coronavirus and at the same time expand capacity in the NHS."
Mr Powis told the programme there had been "outbreaks of altruism and people wanting to help".
He added he had been "bowled over" by medics returning to the front line and the response from people signing up to help the vulnerable.
The Ministry of Defence are helping to set up the hospital, which is expected to have around 4,000 beds in two separate wards.
Housing, Communities and Local Government secretary Robert Jenrick called NHS staff "absolute heroes … working round the clock to support us as a country" and provided more detail on what the volunteers may be doing.
The MP BBC Radio 4's Today Programme that there would be a variety of jobs available including parcel delivery and driving.
Mr Jenrick added: "We need to get volunteers into this collective national effort to support the NHS to support social care, and in particular to support this 1.5 million people who for specific clinical reasons we're shielding."
The Red Cross have praised the NHS using volunteers and pointed towards other organisations also available to help those affected by the illness.
Mike Adamson, chief executive of British Red Cross said the volunteers would be "vital in easing the pressure on our NHS".
He added: "From organisations like the British Red Cross, St John Ambulance and the Salvation Army, to the , there is already an enormous nationwide effort to support people being affected by the coronavirus crisis."
Coronavirus: Everything you need to know