Mr Javid stripped Ms Begum of her British citizenship amid a fierce national debate over whether she should be allowed to return.
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.
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.
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Following news of the boy's death, shadow home secretary Diane Abbott criticised MrJavid's decision.
She tweeted: "It is against international law to make someone stateless, and now an innocent child has died as a result of a British woman being stripped of her citizenship. This is callous and inhumane."
Dal Babu, a former Metropolitan Police chief superintendent and friend of the Begum family, described it as an "entirely avoidable death of a British citizen".
He told BBC's Newsnight: "The family reached out to the Home Office and requested help. The Home Office sent a reply and said 'You've come to the wrong department, you need to speak to Foreign and Commonwealth Office'. There was no attempt to help by the Home Office.
"What we have here is a totally innocent child, whatever you may think of Shamima's shortcomings, the mistakes she made as a 15-year-old child when she was groomed on our watch.
"We failed to safeguard her and now we have failed, as a country, to safeguard a child - a totally innocent British subject."
He added that he believed the decision regarding Ms Begum's citizenship should be "urgently" reviewed.
Mr Javid, when asked whether there was any plan for Ms Begum's son, had previously told the Commons Home Affairs Committee it would be "incredibly difficult" for the Government to facilitate the return of a child from Syria.
"If it is possible somehow for a British child to be brought to a place where there is a British consular presence, the closest place - it might be Turkey for example - in those circumstances I guess potentially it is possible to arrange for some sort of help with the consent of the parent," he added.
Conservative Party chairman Brandon Lewis defended Mr Javid's decision to strip Ms Begum of her British citizenship speaking to BBC Radio 4's Today programme
"The loss of any life of any child is absolutely tragic and is a very clear reminder - this whole case - of the danger of travelling out to that area and getting involved," Mr Lewis said.
"The Home Secretary will have had advice and I know he made a decision based on what is in the national interest and the security of people here in the UK.
"There is no question that the duty of a home secretary in this country is to keep British people safe."
"Inside Syria, whether in a camp or maybe somewhere else, there is no British consular presence."
A Government spokesman said: "The death of any child is tragic and deeply distressing for the family.
"The Foreign and Commonwealth Office has consistently advised against travel to Syria since April 2011.
"The Government will continue to do whatever we can to prevent people from being drawn into terrorism and travelling to dangerous conflict zones."