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.
An eight-year-old boy who was diagnosed with tonsillitis died less than 24 hours after being discharged from hospital.