Tasnime Akunjee tweeted overnight that he has "strong but as yet unconfirmed reports that Shamima Begum's son has died".
It told them that the possibility of bringing the baby to the United Kingdom was a matter for the Foreign and Commonwealth Office and that they would need permission from Ms Begum. He declined to provide further details.
The development comes after British officials ruled out any effort inside Syria to extract Shamima and her child.
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.
Mr Akunjee did not return phone calls and neither the Foreign Office or the Home Office were able to confirm or deny reports.
This 2015 file handout image of a three image combo of stills taken from CCTV issued by the Metropolitan Police show Shamima Begum, centre going through security at Gatwick airport, before they caught their flight to Turkey.
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.
She recently resurfaced in a refugee camp, and gave birth last month.
Begum and her child had been moved from the Al-Hol refugee camp at the start of this month after allegedly receiving death threats from other jihadi brides.
Begum had already lost two other children, and named Jarrah after her first-born son at the request of her terrorist husband Yago Riedijk, who she married in 2015.
Mr Javid has insisted Begum will not be left "stateless" - something which is banned under global law - after her United Kingdom citizenship was stripped, amid speculation that she is a dual British-Bangladeshi national.
Mr Javid also confirmed Ms Begum's son was a British citizen, though he said it would be "incredibly difficult" to facilitate the return of a child from Syria.
He said that the family had wanted to establish communications with Begum to find out what happened to her baby, adding: "This innocent baby was let down".