In an email sent to locum staff, which has been seen by ITV News, a consultant at Northwick Park said staff expect to be in a “terrible position” in the coming weeks and need urgent support.
The doctor says the hospital has seen more than 700 cases pass through in just three weeks.
A separate member of staff at Northwick Hospital has also told ITV News that staff are becoming completely “overwhelmed” on wards and have now started praying before shifts.
It’s the largest daily increase in the UK so far.
The London North West University Healthcare NHS Trust, which runs Northwick Park hospital, is at the centre of the outbreak, with 21 patients who tested positive for Covid 19 having died there over the weekend.
The email sent to locum staff says: “We at Northwick Park have been hit pretty badly by the coronavirus outbreak with probably more than 700 cases which have passed through the hospital in three weeks.
"We expect to be in a terrible position within weeks, with desperately needed staff across the hospital."
"If you would like to help us at NWP then please let me know what you are able to offer.
The Ministry of Defence are helping to set up the temporary NHS Nightingale hospital at the ExCel, while is expected to have around 4,000 beds in two separate wards.
Speaking at the daily Government press conference, Mr Hancock said: "We will, next week, open a new hospital – a temporary hospital – the NHS Nightingale Hospital at the Excel centre in London.
"The NHS Nightingale Hospital will comprise two wards, each of 2,000 people.
A spokeswoman for the London North West University Healthcare NHS Trust said: “Sadly, we can confirm that a further 21 patients who had tested positive for Covid-19 have passed away between Friday 20 March and Monday 23 March, and our thoughts remain with those people who have lost loved ones.
“Our trust is seeing a high number of Intensive Care Unit cases, and we are stepping up our support in response to this demand, having already significantly increased the number of intensive care beds at our hospitals, but we need everyone to also do their bit by staying at home and washing your hands.”
Coronavirus: Everything you need to know