Asked if he supported flying Britons back from Wuhan and elsewhere, Mr Hunt said: "I think I would be very sympathetic and I'm sure the Foreign Office would be too."
He said the NHS is well-equipped to deal with patients returning with the virus but warned that it could strain the service.
"This is a very difficult time of year for the NHS - it is the most difficult time. But, again, my experience is that the NHS does know how to cope with these kinds of emergencies," he said.
"I think the thing that will be difficult is the knock-on impact on other NHS services," he said.
"I think it would be very, very challenging for the NHS in terms of the regular workload but I have absolutely no doubt that, when it comes to doing what comes to necessary to isolate the virus and keep the public safe, our doctors and nurses will do exactly what they need to do."
More than 50 people have now been tested for coronavirus in the UK, according to the Department of Health (DoH), although all tests have returned negative.
As of Sunday afternoon, some 52 people across England, Wales, Scotland and Northern Ireland have been tested for the deadly flu-like virus.
The current risk to the public remains low, the department said, adding that the Government is continuing to monitor the situation closely.
England's chief medical officer Professor Chris Whitty said there was a "fair chance" cases would emerge in Britain as the overall number reported around the world climbed to about 2,744 including 80 deaths, which have all occurred in China.
Britons trapped in the Chinese province at the centre of the outbreak have been urged to leave the area if they are able to do so.
The Foreign Office updated its guidance to "advise against all travel to Hubei province", which has been on lockdown for several days as China seeks to contain the illness.
But the guidance also added: "If you are in this area and able to leave, you should do so. This is due to the ongoing novel coronavirus outbreak."
A pair of British teachers who have been working in Wuhan say they have not left their apartment for five days, that all transport has been stopped and "there is no place to go" and "so we are pretty much stuck".
Jason Neal and Sophie Hunt told BBC Breakfast there has been no reassurance from the British authorities whom they have "struggled" to contact, possibly because of the time difference and them being closed over the weekend. They have about five days of food left and are keeping in touch with colleagues online while the scene outside is now like a "ghost town".
Mr Neal said: "Even if the news is just to sit tight and nothing is going to change - I think it is just the silence that is disconcerting.
"We have not heard anything from outside of Wuhan for a week now."
He said there is a support group for people who may need help and to get masks and some volunteers are going out to make deliveries.
Ms Hunt said emailing and trying to ring the authorities has brought a "useless automated response back from the embassy saying not to go" to the area.
She feels the Chinese authorities have made the right decision by shutting down the city, adding: "All we keep hearing is that the death toll is accelerating every day. All we can do really is sit tight and wait for more news.
"Although it is hard us being stuck here, we know that it is the safest possible option for us at the moment."
Home Secretary Priti Patel said the Government was "looking at all options" to help Britons leave Wuhan following reports that officials have been asked to examine the logistics for an airlift from the city.
One academic told The Guardian his "best guess" was that 100,000 people had been infected with the flu-like virus.
Professor Neil Ferguson, from Imperial College London, told the paper: "There are very large numbers of Chinese tourists across Europe right now.
"Unless the Chinese manage to control this, and I'm sceptical about whether that is possible, we will get cases here."
It comes as spectators celebrated Chinese New Year in central London, which marks the start of the Year of the Rat.
Authorities in China have cancelled a host of events marking New Year as they expand their measures against the virus.
Meanwhile, health officials are continuing to track down around 2,000 people who have recently flown into the UK from Wuhan, the area of China worst affected by the outbreak.
The DoH confirmed it is trying to find "as many passengers as we can" who arrived from the region in the past two weeks to check on their wellbeing.
It is understood Border Force officers have been recruited to help speed up the search for passengers as testing for the virus continues in the UK.
A public health hub has been set up in Heathrow, staffed by a rotating team of seven clinicians working in shifts to support patients on arrival.
One British man, who had travelled to Wuhan to visit his girlfriend, is stuck in the city after his return flight on February 3 was cancelled, and he described trying to get out of the area as "impossible".
The 29-year-old, who did not want to be named, told the PA news agency: "There is no news on when the airport will reopen therefore the airline (China Southern) have just cancelled the flight.
"I've also had no help from the UK Embassy in Beijing who are conveniently closed for the weekend."
Prof Whitty said following a meeting of the Government's Cobra emergency committee in Whitehall on Friday that the virus looked "a lot less dangerous" than contracting Ebola, the recent coronavirus, Mers and "probably less dangerous" than Sars virus.
But he added: "What we don't know is how far it's going to spread, that really is something we need to plan for all eventualities."
"We all agree that the risk to the UK public remains low, but there may well be cases in the UK at some stage."
"We think there's a fair chance we may get some cases over time.
"Of course this depends on whether this continues for a long time, or whether this turns out to be something which is brought under control relatively quickly."