In the letter to Sajid Javid, seen by the BBC, they say they “cannot simply abandon her” and that her status is “a matter for our British courts”.
But they added they were “sickened by the comments she has made” recently.
They also asked for assistance in bringing her newborn baby to the UK.
Ms Begum, who left Bethnal Green, east London in 2015, is living in a refugee camp in northern Syria and gave birth to a son last weekend.
In an interview with the BBC on Monday, she said she did not regret travelling to Syria, though she added that she did not agree with everything the IS group had done.
She told the BBC she was “shocked” by the 2017 Manchester Arena attack – which killed 22 people and was claimed by IS – but she also compared it to military assaults on IS strongholds by coalition forces, saying it was “retaliation”.
The letter, written by her sister Renu Begum on behalf of the family, says: “We wish to make clear, that along with the rest of the country, we are shocked and appalled at the vile comments she has made to the media in recent days.
“These are not…