She has also been told that her British citizenship is to be stripped by the Home Secretary, Savid Javid.
A paramedic told the BBC that the baby had a lung infection and died on Thursday.
Found in a refugee camp in February, an unrepentant Begum sparked a debate in Britain and other European capitals as to whether a teenager with a ISIS fighter's child should be left in a war zone to fend for herself.
Begum went on to make media appearances, shocking Britons, and her family, when she at times defended the Islamic State.
The spokesman said the government had consistently advised against travelling to Syria and would "continue to do whatever we can to prevent people from being drawn into terrorism and travelling to unsafe conflict zones".
Caliph Mirza Masroor Ahmad also acknowledged that taking Ms Begum to the Netherlands with her Dutch husband, Yago Riedijk, was also a possibility.
In an interview with the BBC before authorities had confirmed the baby's death, Javid said: "Sadly there are probably many children, obviously perfectly innocent, who have been born in this war zone". The teen had told newspaper reporters she didn't have a problem with IS actions, including the beheading of captives. "This is a reminder of why it is so, so risky for anyone to be in this war zone".
He added: "Many of us feared this tragic outcome when the Home Secretary washed his hands of Britain's responsibility for a British citizen and a British baby".
Neighbouring Austria said on Wednesday it would not help repatriate any citizens who fought for ISIS and other militant groups, as countries across the West wrestle with how to deal with returning militants. However, the Home Office noted, that agency has not had consular representation in Syria since 2012.
Begum's family said the boy appeared to be in good health when he was born on February 17.
It was reported that she could be eligible for Bangladeshi citizenship, but Dhaka said there was "no question" of her being allowed to enter the country.
That puts Begum at risk of statelessness - a status that United Kingdom law is set up to avoid.
Two senior members of the government said on Saturday that the death was a tragedy but that the home secretary took the decision on grounds of national security.