Separately, a paramedic told the BBC that Jarrah, the almost three-weeks-old baby, died on Thursday after battling a lung infection.
The Begum family's lawyer, Tasnime Akunjee, said he had received confirmation that the child, who was only a few weeks old, had died and that Begum's relatives were devastated upon hearing the news.
A reporter for the BBC spoke to a paramedic working for the Kurdish Red Crescent who said that the baby...
Home Secretary Sajid Javid stripped the British citizenship of Ms Begum - who is from Bethnal Green in east London - amid a fierce national debate over whether she should be allowed to return.
Now 19, Begum told journalists she wanted to raise Jarrah in the United Kingdom, alleging she had lost two other children in Syria to malnutrition and disease.
Javid previously confirmed the baby was a British citizen and said he had considered the child's interest when deciding to revoke Begum's citizenship.
Ms Begum's British citizenship was revoked on the grounds that she is eligible for citizenship of Bangladesh until the age of 21 through her parents' Bangladeshi dual nationality.
She was 19 when she gave birth to her baby, Jarrah, in a Syrian refugee camp in February.
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Mustafa Bali was quoted as saying by news agencies on Friday that Shamima Begum's infant had died at a camp in north Syria.
Her family can challenge the decision before the Special Immigration Appeals Commission - Begum's reported Bangladeshi status is expected to be a prime focus as part of that process.
Shamima Begum speaking to the BBC on 18 February.
She resurfaced heavily-pregnant a refugee camp in northern Syria last month and spoke of her desire to return to the United Kingdom, as the self-styled caliphate collapsed.
Ms Begum said she had previously lost two other children and named her new son Jarrah after her firstborn.
"This is a reminder of why it is so, so unsafe for anyone to be in this war zone", the UK's Home Secretary Sajid Javid told the BBC upon news of the boy's death.
Begum's lawyer Mr Akunjee said: "I confirm that it is our understanding that Shamima has been moved from Al-Hol due to safety concerns around her and her baby".
"Inside Syria, whether in a camp or maybe somewhere else, there is no British consular presence".