A medical source told the BBC the two-week-old baby had contracted a lung infection and died on Thursday.
Mohammed Tasnime Akunjee said on Twitter that he had received "strong" but "unconfirmed" reports that Miss Begum's baby boy, who was born barely a few weeks ago, has died.
She was 15 when she and two other schoolgirls from Bethnal Green, east London, went to join IS in February 2015.
It was in reaction to the minister's decision to revoke 19-year-old Begum's British citizenship on security grounds, indicating that she would have the right to acquire Bangladeshi citizenship by virtue of her parents' heritage.
A then-heavily pregnant Begum told The Times she was concerned for the safety of her unborn child as she had already lost two children during her 4 years with the militant group.
She gave birth shortly afterwards and said she wanted to return to the United Kingdom, but the Home Office stripped her of her British citizenship. However, the Bangladesh foreign ministry said in a statement that Begum is not a Bangladeshi citizen, nor has she ever visited the country.
The paramedic said both Ms Begum and her baby were taken back to the camp for burial.
Shamima Begum captured on CCTV in 2015 on her way to the Middle East. Credit PA
Begum, 19, gave birth to her third child Jarrah in mid-February, while living at a camp run by the Syrian Democratic Forces (SDF), a US -backed alliance of Kurds and other groups that is working to drive the Islamic State out of the last territory they hold.
She married a Dutch man within days of her arrival in Syria when she was 15.
Begum's son is one of scores of children who have died fleeing from fighting in ISIS' last remaining Syrian enclave Baghouz. The family has said it plans to challenge Javid's decision.
Begum's family had announced the boy's birth on February 17, saying they believed he was "in good health".
Before she gave birth, Begum said she wished to bring up her baby in the United Kingdom, and her family have begged for her to be shown mercy and to be allowed to return to east London.
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.
'Inside Syria, whether in a camp or maybe somewhere else, there is no British consular presence'.
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