Tasnime Akunjee tweeted: "We have strong but as yet unconfirmed reports that Shamima Begum's son has died. He was a British Citizen."
Shamima Begum, 19, was one of three schoolgirls, along with Kadiza Sultana and Amira Abase, from Bethnal Green Academy who left the UK to travel to Syria in February 2015.
Before she gave birth, Begum said she wished to bring up her baby in the UK, and her family have begged for her to be shown mercy and to be allowed to return to east London.
Their lawyer Aamar Anwar told the Sunday Mirror: "If it was your daughter would you not want her back? Aqsa's mother and father vociferously condemned their daughter."
Mr Rafiq, formerly a member of a Government task force looking at countering extremism after the 2005 terrorist bombings in London, told the Press Association: "Nobody can tell whether at this stage, without meeting her, spending time with her, whether she could be de-radicalised or not."