Begum was 15 when she fled east London to travel to the war-torn country to marry an Islamic State fighter in 2015.
Now 19, Begum told journalists she wanted to raise Jarrah in the United Kingdom, alleging she had lost two other children in Syria to malnutrition and disease.
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".
Tasnime Akunjee tweeted: "We have strong but as yet unconfirmed reports that Shamima Begum's son has died".
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. "He was a British citizen".
She has given a number of interviews to British media, saying she wanted to return to Britain.
"I don't really want to stay here, I don't want to take care of my child in this camp because I'm afraid he might even die in this camp", she said in an interview with United Kingdom broadcaster Sky News shortly after giving birth.
Ms Begum recently resurfaced in a refugee camp, and gave birth last month.
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.
He added: "I know (Shamima) is not a danger to anyone whatsoever. but she made a mistake and will have to live with the consequences.
I wish I had stayed low and found a different way to contact my family", she said.
She sparked a huge debate in the United Kingdom over whether she should be allowed back.
Last week, Akujnee said: "I can confirm that it is our understanding that Shamima has been moved from Al-Hol due to safety concerns around her and her baby".
Earlier this week, Mr Akunjee tweeted a screenshot of the reply that they had received from the Home Office.
The FCO is obliged to consider requests for consular assistance, the letter added.