The tests will provide an "early warning" if the virus is spreading, Public Health England's Prof Paul Cosford said.
It comes as oil firm Chevron asked 300 London staff to work from home while one employee is tested for the virus.
And more schools closed after trips to Italy, which has more than 300 cases.
Italy has in recent days become Europe's worst-affected country, with a surge in cases that appear to have spread to Austria, Croatia and Switzerland.
Health Secretary Matt Hancock told the House of Commons that it was important not to overreact in response to the outbreak, saying that would also have "economic and social" costs.
He also said UK employers had been sent guidance telling them staff who are asked to self-isolate are entitled to take sick leave.
Mr Hancock said it was "a very important message for employers" that those who need to self-isolate can do so "as if they were sick".
Who will be tested?
Up to now, people have only been tested if they displayed symptoms having recently returned from one of the countries where there has been an outbreak, including China, South Korea and northern Italy.
Public Health England said it was now working with some hospitals and GP surgeries to conduct tests on some other patients.
In eight hospitals, patients in intensive care with severe respiratory infections will be tested for the virus.
In 100 GP surgeries, those coming in with milder flu-like symptoms - dry coughs, fever, shortness of breath - will be tested.
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Prof Paul Cosford, emeritus medical director at Public Health England, told BBC Radio 4's Today programme that it was "heightening our vigilance" because of the apparent spread of the virus in countries outside mainland China.
The eight hospital trusts involved are Brighton and Sussex University Hospitals, Guy's and St Thomas', Royal Brompton and Harefield, Royal Papworth Hospital, University Hospitals of Leicester, University Hospitals of South Manchester , Nottingham University Hospital and Sheffield Teaching Hospitals.
The plan is expected to mean hundreds more people are tested for the virus each week. There have been 7,132 tests carried out in the UK since the outbreak began to spread beyond China in January.
Of these, 13 were confirmed positive - including four people who returned from the Diamond Princess cruise ship in Japan.
How have schools been affected?
Several schools have sent pupils home or closed after they returned from skiing trips in northern Italy over half term after the government updated the advice for travellers returning from Italy.
Hugh Hegarty, chief executive of the trust which runs Trinity Catholic College in Middlesbrough, said the school closed because of the "potential risk" after students and staff returned from a ski trip near Verona.
He said they advised children who had been on the trip to self-isolate at home, while the school as a whole would close for 72 hours, during which there would be a deep clean and school leaders would monitor the situation.
Among the other schools affected are:
Tudor Grange Academy, Kingshurst, in Birmingham
Ysgol Friars in Bangor
Lutton St Nicholas Primary School and Gedney Church End Primary School in Lincolnshire
St Aidan's High School in Harrogate
St Christopher's C of E High School in Accrington
Three schools in Northern Ireland - Limavady Grammar School, Banbridge Academy and Cambridge House Grammar in Ballymena
All Hallows Catholic School, in Farnham, Surrey
Public Health England said it was not giving "blanket advice" that schools should close if staff or pupils had travelled to areas with outbreaks of the virus.
How have travel plans been hit?
British Airways has cancelled some of its flights to Milan because the
coronavirus outbreak had resulted in "reduced demand", it said.
The airline said 22 round-trips between London and Milan had been cancelled between 27 February and 11 March, adding that customers could get refunds or book for a later date.
BA has also offered passengers travelling to or from other areas of northern Italy - Turin, Bologna, Venice, Bergamo and Verona - up to 2 March the chance to rebook.
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British citizens are among hundreds of guests confined to a hotel in Tenerife after a visiting Italian doctor tested positive for the infection.
The Foreign Office said it was offering advice and support to a number of people - there are understood to be 168 Britons staying in the hotel.
According to health officials in the Canary Islands, more than 100 guests could be released in the coming hours, but it was not clear if any Britons were included.
The remaining 600 or so guests will need to remain quarantined for a fortnight, officials said.
One guest, Rosie Mitford, told the BBC's Victoria Derbyshire programme they were being allowed out in the grounds once their temperatures had been tested.
She said: "They've given us a mask and you can wander about if you want, but really you should stay in your room. Obviously it's an awful situation. Everyone's just keeping their spirits up as much as they can, really."
Today programme presenter Nick Robinson has recounted his experience of being tested for the
coronavirus, after returning from holiday in Vietnam and Cambodia.
Mr Robinson said he was feeling "fine" as he awaited the results in self-isolation at home, although he was a "little bit croaky" because he had a cough before he left the UK.
He said he was instructed to drive himself to hospital if possible, to avoid the risk of infecting others. A nurse came out into the car park wearing a plastic face visor, rubber gloves and a disposable apron.
The nurse carried out tests for temperature, blood pressure and oxygen saturation through his car window, before taking a swab for the
coronavirus test.
When they wanted a second opinion from a doctor, he was walked through a staff car park through a back entrance, again to reduce the risk of infecting other patients.
"It's the sheer scale of the effort that occurs to you. I was there for more than two-and-a-half hours," he said. "This is hugely time consuming."
