Callum Cartlidge suffered a cardiac arrest at home after developing suspected sepsis, sources told the BBC.
Paramedics were allegedly told to take him 18 miles (29km) to Worcestershire Hospital and not nearby Alexandra Hospital in Redditch, which stopped admitting children to A&E in September.
The trust said it was investigating.
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Callum, from Redditch, had seen a GP on 28 February and was diagnosed with tonsillitis and a tummy upset and given antibiotics.
His mother Stacey said her son got worse and on 2 March a GP sent him to Worcestershire hospital.
She said he was discharged at 23:00 GMT and she was told to give him Calpol. He collapsed the next day and later died.
West Midlands Ambulance Service said the trust had previously told staff not to take patients under 16 to Redditch, after Alexandra Hospital's children's inpatient services were moved.
Sources claim paramedics, who suspected Callum had developed sepsis, wanted to take him to Redditch but were told they could not.
Three members of staff performed life support throughout the 23-minute journey to the hospital.
"Having reviewed the case, there is no doubt that the ambulance staff did absolutely everything possible to help the child," a service spokesman said.
Worcestershire Acute Hospitals NHS Trust offered its condolences to Callum's family.
"As with any sudden unexpected death in childhood there will be a referral to the coroner, a post-mortem [examination] and a full external Sudden Unexpected Death in Infants And Children investigation, as well as our own internal processes for a serious incident investigation," a spokesperson said.

The government said any decision should await a report being written by the former bishop of Liverpool James Jones.
Andy Burnham, who called for the law, criticised the failure to include it in the Policing and Crime Bill.
He said it was wrong Margaret Aspinall had to use compensation for her son's death towards the "fight for justice".
The shadow home secretary said Ms Aspinall, whose son James died in the 1989 disaster, had to put the Â£1,000 she received in compensation towards legal fees and was then "living on the breadline" trying to cover the cost of fighting for justice.
The Leigh MP said the 27-year fight Hillsborough families had to endure showed how the odds were stacked against families seeking the truth.
It has been revealed South Yorkshire Police spent at least Â£2.1m on legal fees representing its suspended Chief Constable David Crompton during the inquests into the deaths of the 96 fans who were killed.
The Labour MP said: "This Bill was an opportunity to make this country fairer, to even up the scales and tip them in favour of ordinary families and away from the establishment.
"I fear we have failed to do that."
He said Ms Aspinall, who is chairwoman of the Hillsborough Family Support Group, watched the debate from the public gallery, and was "disappointed" by the government's refusal to accept equal legal funding in to the Bill.
The Policing and Crime Bill gives police stronger powers and hands responsibility for fire and rescue services to elected police and crime commissioners.
It creates so-called super complaints which will allow organisations like charities to bring a large-scale complaint against the police and strengthens protections for whistleblowers so identities are protected while investigations are conducted.
Home Secretary Theresa May said the Bill would allow police to go "further and faster" and "radically reform" the complaints and disciplinary systems.
However, Mr Burnham said it was a "mixed bag" that included improvements but also missed opportunities including the equal legal funding for victims as well as a second Leveson Inquiry into the relationship between police and the press.
The Bill will now pass to the House of Lords for further scrutiny.
In April, jurors ruled the 96 victims of Hillsborough were unlawfully killed.
The Rt Rev James Jones, the former Bishop of Liverpool, led the Hillsborough Independent Panel now acts as the home secretary's adviser on Hillsborough.
An eight-year-old boy who was diagnosed with tonsillitis died less than 24 hours after being discharged from hospital.
[NEXT_CONCEPT]
A decision on a "Hillsborough Law" to ensure legal funding for bereaved families at inquests where police are involved has been put on hold.