The letter says: "We have a duty to her, and a duty to hope that as she was groomed into what she has become, she can equally be helped back into the sister I knew, and daughter my parents bore".
Her family's lawyer, Tasnime Akunjee, is looking at challenging Britain's decision to revoke her citizenship, which risks leaving Begum stateless.
The British-born 19-year-old, who travelled from east London to Syria to join Islamic State in 2015, wants to return from Syria because her newborn son is unwell, and she does not wish to allow him to return to the United Kingdom alone.
An LBC listener broke down in tears as she defended Shamima Begum, saying the Isis bride was "forced" to go to Syria under "false pretences".
As the baby named Jarrah was born while Begum was still a British national last week, his status remains that of a British national.
However, her family are exploring legal and practical options to bring her baby to London while she undergoes what could be a long drawn-out appeal against the removal of her British citizenship.
This undated photo issued by the Metropolitan Police shows Shamima Begum, who had left the United Kingdom to join the Islamic State.
"We do not comment on individual cases, but any decisions to deprive individuals of their citizenship are based on all available evidence and not taken lightly", a spokesperson said.More news: SpaceX rocket launch: Falcon 9 spacecraft take-off from Cape Canaveral
Many have argued the teen should not be allowed to return, while others say she should face prosecution and attempts should be made to deradicalise her.
The UK home secretary's own counter-terrorism adviser, Sara Khan, also branded the move "counterproductive".
Muthana's case is different than Begum's, however, because USA officials argue that she was never a citizen in the first place - a premise denied by her family.
However, Dhaka has since distanced itself from such a prospect, categorically denying Shamima had any claim to a Bangladeshi nationality or the right to entry to that country.
He suggested the Home Office, headed by home secretary Sajid Javid, has made the decision for the good of the country.
"If she at least admitted she made a mistake then I would feel sorry for her and other people would feel sorry for her".
We contacted and cooperated with all the relevant government agencies in both the United Kingdom and Turkey to try and stop her progress.
"Taking somebody's citizenship away is not the right thing to do".
He also said that she "had no regrets, she was calm and composed but she was also in a state of shock - she had just come out of a battlefield, nine months pregnant, many of her friends dead through airstrikes and all the rest of it - so I wouldn't want to rush to judge her too harshly".