An investigation has been launched after a whistleblower claimed that medics working for the NHS 111 had been so overworked they had fallen asleep while on duty.
Sarah Hayes, a former senior call adviser for the helpline in the South West, said she had to speak out as she felt the service was "unsafe".
Ms Hayes worked at the same service as a call handler who failed to recognise that one-year-old William Mead, who died following a string of NHS failings, had sepsis caused by an underlying chest infection.
Writing in the Daily Mail, Ms Hayes said: "The service at the moment is unsafe, particularly for children and babies. I don't want more children put at risk like William was."
She said there were "frequently" no on-call clinicians in the call centre, run by South Western Ambulance Service NHS Foundation Trust, while "overworked" paramedics and nurses fell asleep on their shifts.
The baby's mother, Melissa Mead, said the comments by Ms Hayes represented a "very chaotic system" that allowed for compromises in care.
Ken Wenman, chief executive of South Western Ambulance Service NHS Foundation Trust, insisted that patient care and safety were "top priorities" for the organisation.
He said the trust had worked personally with the families involved in cases referred to in the Daily Mail "wherever possible" to address their concerns and prevent such incidents from happening again.
"We have issued apologies and assurances to those families, and would like to take this opportunity to thank them for their input," he said.
"We pride ourselves in being open and transparent, and have not misled anyone.
"There are a number of allegations made in the Daily Mail that we strongly refute; however, there are actions that Sarah Hayes says she took, for which we can find no paper trail or audit, and therefore an investigation into these allegations has been commissioned.
"We are proud of the work that our staff deliver day in, day out, and are fully confident of the robust procedures we have in place around the NHS 111 service."
He added that the trust was "surprised and disappointed" that Ms Hayes had not followed its whistleblowing process and that she had taken "so long" to make the allegations.
In the Daily Mail article, Ms Hayes said some callers would wait more than 12 hours for a call back from a medically-trained member of staff and at night up to 65 patients could be on the waiting list.
"The nurses and paramedics we did have were so exhausted and overworked that some would fall asleep on shift. I was angry, of course, but I don't feel it was their fault. Put simply, they were exhausted," she said.
"I think anyone with experience of 111 would say it has problems for young babies, and it's really hard to get a good assessment done.
"I think that passing a young baby to a clinician would be a really good idea but you would need many, many more clinicians to make it work."
Following the allegations, Mrs Mead, from Penryn, Cornwall, said: "I think everything that is being presented represents a very chaotic system and when there is a chaotic system it allows for compromise in care.
"I didn't know that William was seriously ill - I had already been to one doctor who had sent us away and so I was looking for reassurance.
"When I spoke to that call handler I expected him to be asking the appropriate questions, I expected him to be listening, I expected him to be putting the answers in correctly and making the right judgments, and when he made the wrong judgment, albeit for whatever reason, I then got signposted in the wrong direction.
"With the benefit of hindsight I know that but at the time I was reassured and thought 'OK that's fine'."
She said she did not wish to comment on the pictures of medics allegedly asleep at their desks but added that NHS Trusts need to take responsibility for their staff.
"It's the trusts that need to make sure they employ enough staff to cover the volume of calls," she added.
"There needs to be more clinicians and nurses available for people to be able to hand those calls over where clinically necessary."
Labour's shadow health secretary Heidi Alexander said: "These reports are deeply concerning and once again show a service that is simply not fit for purpose.
"Patients should have confidence that when they call NHS 111 they will get medical advice from a trained healthcare professional. However, these revelations raise serious questions about the quality of the current service and its ability to spot patients with life-threatening illnesses.
"Jeremy Hunt needs to urgently review this new evidence and guarantee patients that the problems besetting NHS 111 will be addressed."
Last month, the Health Secretary apologised to the Mead family, saying they were let down in the "worst possible way" after details of a string of NHS failings emerged in a report.
According to the report by NHS England, Mrs Mead spoke to medics at least nine times in the 11 weeks leading up to William's death in December 2014.
He was seen by several GPs who failed to spot that his condition was deteriorating.
On the day before his death, Mrs Mead called 111 for advice and also spoke to an out-of-hours GP who did not have access to any of her son's medical records.
The 111 call handler failed to explore further some of Mrs Mead's comments about William's condition, including that his temperature had gone from a high 40C (104F) to a low 35C (95F) - a sign of sepsis.
But the report also blamed GPs for the baby's death, saying a "significant missed opportunity was the fact that the underlying pathology, a chest infection and the pneumonia in the last six to eight weeks or so of William's life, were not recognised and treated".