The baby died later that day.
A British government spokesman said: "The death of any child is tragic and deeply distressing for the family".
A medical source speaking to the BBC the boy was rushed to hospital Thursday morning with breathing difficulties and a lung infection.
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.
Ms Begum was 15 when she and two other schoolgirls went to join the terror group in February 2015.
He said he believed she had been moved to another camp near Syria's border with Iraq.
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The judge himself even noted that when Manafort addressed the court Thursday, he didn't express any regret for his crimes. Among the things that will be decided is whether the D.C. and Virginia sentences will run consecutively or concurrently.
Begum is British-Bangladeshi, according to reports, but Bangladesh said she's not a national of that country and won't accept her there. She named him after her first son, who died before she reached Syria's Al-Hawl refugee camp.
Earlier on Friday, the lawyer representing the family of Ms Begum, Tasnime Akunjee, said he had "strong but unconfirmed reports" that the baby had died.
As her child was born before she was deprived of United Kingdom citizenship, the baby would still be considered British.
Begum told ITV that it was "kind of heartbreaking to read" a copy of the letter, calling the move "hard to swallow" and "a bit unjust on me and my son".
Ms Begum's parents are from Bangladesh but her family said she is not a dual citizen.
Her family can challenge the decision before the Special Immigration Appeals Commission - Begum's reported Bangladeshi status is expected to be a prime focus as part of that process.
They also pointed out that Shamima Begum does not have the relevant paperwork to reside there.
The Sun reported that Shamima, who wants to return to the United Kingdom but has been stripped of her British citizenship, had received death threats since speaking out about her plight. In order to protect this country, he has the power to deprive someone of their British citizenship where it would not render them stateless.