Begum, who left east London to join ISIS in Syria when she was 15, made worldwide headlines last month as she publicly pleaded with the United Kingdom government to be allowed to return. The baby died a few hours after arriving at the hospital, the NGO said.
Last week it was reported that Shamima and her son Jerah, named after a 7th century Islamic warlord, had been moved from the refugee camp where he was born, after they were "threatened".
Home Secretary Said Avid said he did not know if the baby, a British citizen, had died.
The paramedic, working for the Kurdish Red Crescent in and around the camp where Ms Begum had been staying, said the baby had been suffering from breathing difficulties and a lung infection.
"He was a British Citizen".
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Ms Begum was 15 when she and two other schoolgirls went to join the terror group in February 2015, and married Yago Riedijk, now 27, shortly afterwards.
On February 17, her family announced the boy's birth and said they believed he was "in good health".
"Shamima Begum returned to the internment camp and her son was buried there".
Earlier this week, Mr Akanke tweeted a screenshot of the reply that they had received from the Home Office. Now 19, she has already lost two children prior to her latest child being born.
"These matters should be investigated", the 60-year-old said, urging the United Kingdom to allow his daughter to return and face the British legal system.
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She wanted to return to Britain but was stripped of her citizenship.
The SDF, the rebel group backed by the United States that controls the camp where Ms Begum, 19, is living, had earlier insisted that the baby was alive after a lawyer for Ms Begum's family, Tasnime Akunjee, tweeted that he had "strong but unconfirmed reports" that Jarrah had died. The family said it planned to challenge Mr Javid's decision. In a letter obtained by the BBC, Renu called her sister's son "the one true innocent", adding that he "should not lose the privilege of being raised in the safety of this country".
Ms Begum's British citizenship was revoked on the grounds that she is eligible for citizenship of Bangladesh until the age of 21 through her parents' Bangladeshi dual nationality.
Under global law, the United Kingdom can revoke a citizenship of a British national only if the individual would not be made stateless.
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