Tasnime Akunjee said in a statement: "We have strong but as yet unconfirmed reports that Shamima Begums son has died".
Last month, Mr Javid confirmed the boy was a British citizen and said he had considered the child's interest when deciding to revoke Ms Begum's citizenship.
Shamima Begum, 19, was one of three schoolgirls, along with Kadiza Sultana and Amira Abase, from Bethnal Green Academy who left the United Kingdom to travel to Syria in February 2015.
However, it had been reported that she had fled another refugee camp recently after threats of reprisals against her after she spoke about her plight.
Their lawyer Aamar Anwar told the Sunday Mirror: "If it was your daughter would you not want her back?"
The news of his possible death follows a plea from Ms Begum's London-based family to United Kingdom home secretary Sajid Javid to assist them in bringing the baby back to the United Kingdom based on his rights as a British citizen.
Shamima Begum, 19, gave birth to her son Jarrah in a refugee camp after fleeing ISIS and was reportedly moved to a detention camp's hospital "under armed escort" on Thursday.
The paramedic, working for the Kurdish Red Crescent in and around the camp where Ms Begum had been staying, said the baby had been suffering from breathing difficulties and a lung infection.
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Begum told reporters that she had lost two other children to malnutrition and disease.
Ms Begum's British citizenship was revoked on the grounds that she is eligible for citizenship of Bangladesh until the age of 21 through her parents' Bangladeshi dual nationality.
After the Bangladeshi government claimed she was not a Bangladeshi citizen, she said she had no desire to go to the country.
Begum had moaned after being banned from Britain, telling ITV: "I'm a bit shocked".
She added: "It's kind of heart-breaking to read".
Her family, who pledged to appeal against the decision, also wrote to Mr Javid pleading with him to allow a safe passage for the boy to come to the UK.
Labour leader Jeremy Corbyn was among those backing her bid to be allowed back to the UK.
Ms Begum, who left the United Kingdom in 2015, was found in a Syrian refugee camp in mid-February.
On February 17, her family announced the boy's birth and said they believed he was "in good health".