The community testing regime - expanded to cover all 317 local authorities - uses rapid lateral flow tests, which can return results in 30 minutes.
Local councils are being encouraged to prioritise tests for those who cannot work from home during the lockdown.
Meanwhile, thousands of people over 80 are being invited to book vaccinations.
NHS England says 130,000 letters with invitations for vaccinations at one of seven new regional centres began arriving on doormats this weekend, with more than 500,000 following this week.
On Saturday the number of people who have died in the UK within 28 days of a positive Covid test reached 80,000.
Scientists advising the government have warned that lockdown measures in England need to be stricter to achieve the same impact as the March shutdown.
England's chief medical Prof Chris Whitty said that, unless people started to follow the rules more strictly, emergency patients will have to be turned away from hospitals, causing "avoidable deaths".
Writing in the Sunday Times, Prof Whitty said: "Every unnecessary interaction you have could be the link in a chain of transmission which has a vulnerable person at the end."
Meanwhile, the chair of the British Medical Association, Chaand Nagpaul, revealed that more than 46,000 hospital staff are off sick with Covid-19, which is adding to the pressure on the NHS and beginning to affect the vaccination programme, the Observer reports.
Under the national lockdown, people in England must stay at home and can go out only for limited reasons such as food shopping, exercise, or work if they cannot do so from home. Similar measures are in place across much of Scotland, Wales and Northern Ireland.
The Department of Health and Social Care said expanding the Community Testing Programme to more people without symptoms was "crucial given that around one in three people" who contract Covid-19 show no symptoms.
It said regular community testing using the rapid tests had already identified more than 14,800 positive Covid-19 cases.
So far, 131 local authorities in England have enrolled in the government's community testing programme, with Milton Keynes, Slough, Doncaster and Essex the latest to join.
Health Secretary Matt Hancock said targeted asymptomatic testing and subsequent isolation was "highly effective in breaking chains of transmission".
"Lateral flow tests have already been hugely successful in finding positive cases quickly - and every positive case found is helping to stop the spread - so I encourage employers and workers to take this offer up," he said.
NHS Test and Trace will also work closely with other government departments to scale-up workforce testing, the Department of Health and Social Care said.
Many are already piloting regular workforce testing, with 15 large employers having taken up this offer already across 64 sites, "including organisations operating in the food, manufacturing, energy and retail sectors, and within the public sector including job centres, transport networks and the military".
An estimated 27,000 tests have taken place across the public sector as part of pilots so far, the department added.
The Department of Health and Social Care said plans were already in place for rapid testing of staff and students in schools and colleges and staff in primary schools.
It said primary schools would start to get home testing kits for staff shortly to test people on a weekly basis, while "secondary schools and colleges have set up test sites, and have started testing staff and students present on site during the lockdown".
Business has welcomed the move.
Matthew Fell, of the Confederation of British Industry (CBI), which represents 190,000 UK businesses, said rapid mass testing was "a key piece of the puzzle when it comes to getting on top of the virus".
He said health and safety of staff was paramount, adding: "This expansion of testing will help more critical workers and those unable to work from home to operate safely, while also catching new cases more swiftly."
Meanwhile, London Mayor Sadiq Khan has welcomed the opening of a vaccination centre at the ExCel Centre in east London next week - which is also home to one of the Nightingale hospitals.
He said it would be crucial in helping to accelerate the rollout of the vaccine, adding that the capital was in a "critical" situation because of the threat of the virus.
The other six England centres include Ashton Gate in Bristol, Epsom racecourse in Surrey, Newcastle's Centre for Life, the Manchester Tennis and Football Centre, Robertson House in Stevenage and Birmingham's Millennium Point.
Elsewhere, the newly approved Oxford/AstraZeneca vaccine will be available in more than 1,000 locations in Scotland from Monday, the nation's government has said.
Buckingham Palace said the Queen, 94, and the Duke of Edinburgh, 99, received Covid-19 vaccinations on Saturday.
They are among about 1.5 million people in the UK to have had at least one dose of a Covid vaccine so far.
On Saturday, 12 people were arrested during an anti-lockdown protest in south London.
Elsewhere, in Wales, police have said people were "blatantly" ignoring rules on lockdown restrictions despite repeated warnings after more than 100 cars were turned away from a beauty spot on Saturday morning.