Ms Begum was a schoolgirl when she left Bethnal Green in east London in 2015.
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 USA -backed alliance of Kurds and other groups that is working to drive the Islamic State out of the last territory they hold. "He was a British Citizen", Mr Akunjee wrote.
Begum's child, Jarrah, passed away following what appeared to be a lung infection, the BBC reported a local paramedic as saying.
A fierce national debate erupted on whether Begum should be allowed to return to the United Kingdom as Home Secretary Sajid Javid revoked her British citizenship.
Ms Begum was 15 when she caught a flight in 2015 to join IS.
She gave birth to her newborn boy last month.
Begum is British-Bangladeshi, according to reports, but Bangladesh said she's not a national of that country and won't accept her there. Almost 100 children have died en route or shortly after arriving at al-Hawl refugee camp in northern Syria due to a lack of food, water and health care, according to the International Rescue Committee, which says the situation there has reached a "breaking point".
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On Friday, the Begum family lawyer Tasnime Akunjee tweeted: "We have strong but as yet unconfirmed reports that Shamima Begum's son has died".
She sparked a huge debate in the United Kingdom over whether she should be allowed back.
The letter asked Begum's family to make her aware of the decision and of her right to appeal.
On Tuesday 19th February, a Home Office spokesperson said: "In recent days, the home secretary has clearly stated that his priority is the safety and security of Britain and the people who live here".
Renu, eldest sister of Shamima Begum holds her sister's photo whilst being interviewed by the media at New Scotland Yard, Central London.
A child born to a British parent before they are deprived of their citizenship would still be considered British.
Speaking from al-Hawl before giving birth, Begum told United Kingdom newspaper The Times that she wanted to come home to have her child.
Mr Javid made the decision after receiving advice that Ms Begum could qualify for Bangladeshi citizenship.