Telephone and internet advice service NHS Direct has recorded a 50% increase in calls, due to illnesses linked to the cold weather.
The organisation received about 46,000 calls last weekend and is hiring more staff to deal with the number of inquiries.
The Daily Mail newspaper said the service was "at breaking point" with long waiting times for medical advice.
But the Department of Health said the service was coping "very well".
NHS Direct's chief executive, Nick Chapman, has apologised to patients who have had to wait longer than expected to be dealt with.
He said: "The excessively cold weather creates demand for health care generally, and in particular for telephone services that you do not need to leave the house to access.
"We are taking a number of actions to address and improve the service at this busy time, which include increasing the number of permanent and temporary staff and providing more opportunities for staff to work from home for short periods at our busiest times."
The Mail said almost 960 people an hour called the helpline last weekend, with the senior nurse telling the paper that the situation was "by far the worst it has ever been".
The paper claimed some people had had to wait for up to two days to speak to a nurse, after making the initial call.
NHS Direct said that for the week commencing 13 December, 99% of urgent calls were dealt with within 20 minutes.
It also said 59% of less urgent calls were dealt with in an hour, with 90% being dealt with in four hours.
And 69% of non urgent calls were dealt with within two hours, with 90% being dealt with during the same day.
The Department of Health announced on Tuesday that 302 people were currently in intensive care with flu, and that 17 people had died.
A Department of Health spokeswoman said the NHS is "well prepared".
She said: "We are currently seeing an unusually high number of people in critical care with flu.
"There is always more pressure on the NHS at this time of year and this year is no different.
"But the NHS is coping very well with only a small percentage of the intensive care capacity being taken up by patients with flu.
"However, as a cautionary approach, local health trusts are looking at how they can increase capacity if necessary."