Up to a million children at risk from serious illnesses are being urged to have a flu jab.
Different types of flu virus are prevalent each year
Health officials said fewer than half of those with conditions such as chronic heart or chest complaints had been vaccinated.
But they say there is no need for a mass vaccination of children.
Around half a dozen children aged between 18 months and 16 are thought to have died from the Fujian strain of influenza this winter.
The latest victim of the strain in the UK is thought to be three-year-old Tamara Whitty, from Warndon, in Worcester, who died on Friday.
But an inquest has revealed that 12-year-old girl Fern Summers from West Newton in Norfolk, who it was feared had died after contracting the flu strain, actually died after an asthma attack.
Those aged over 65 and "high risk" adults, such as those with asthma are already offered a free flu vaccination under a national scheme.
This year's vaccine is not specifically targeted at the Fujian strain, but experts say it does offer some protection.
'Step into the unknown'
Sir Liam, Chief Medical Officer for England, said expert advice from the Joint Committee on Vaccination and Immunisation (JCVI), suggested there was no need to extend the current vaccination programme to children.
But he said the committee had recommended more young people in high-risk groups should have the jab.
He added: "We have seen a small number of deaths in children this year.
"These are obviously tragic when they occur but the level of deaths in children isn't different to previous years.
"To take a step into the unknown and vaccinate children when we are not sure of the effect of vaccination wouldn't be indicated."
Children under four appear to be most at risk of the latest
strain of flu, which originated in China.
In the week ending November 23, 128 children under four per 100,000 visited their GP with flu-like symptoms, compared to 56 visits for five to 14-year-olds.
Among the over-65s, there were just 40 visits related to flu per 100,000 people, according to figures from the Royal College of General Practitioners.
An outbreak is not considered to be an epidemic until there are 400 cases per 100,000 people.
'Lack of experience'
Sir Liam said the Fujian strain did appear to particularly affect children, perhaps because older people have some protection from previous flu jabs.
He said cases of flu and deaths so far this year had been slightly above average but were not too significant.
"After three quiet years flu is appearing and appearing earlier than in the past.
"We can't predict the eventual size of this year's outbreak but so far there has not been a major surge. We could always be surprised."
John Oxford, Professor of Virology at Queen Mary's School of Medicine in London, said children, together with the elderly and those suffering diabetes, severe asthma and heart disease, were particularly vulnerable to the "virulent" Fujian strain.
He said: "The reason it affects the young is probably because there's not been much flu in Britain in the last two to three years.
"Younger people haven't experienced much flu. They are much more open to catching it because they haven't been able to build up any natural immunity."