Lola and Isabella Koupparis, aged nine months, were found crying and covered in blood after the fox went into their upstairs bedroom and attacked them in their cots in Hackney, east London.
They were taken to the Royal London Hospital in east London after the incident on Saturday night but Isabella was transferred to Great Ormond Street Hospital in central London and is likely to be scarred for life.
The Royal London Hospital said Lola remained in a serious but stable condition.
Meanwhile, London Mayor Boris Johnson said foxes were a pest and a menace and could in rare circumstances pose a threat to humans. He said: "Therefore it's right that boroughs should focus on their duties for pest control because as romantic and cuddly as a fox is, it is also a pest."
Family members reportedly said that the fox attack could be "life-changing" for the twins.
Their mother, Pauline Koupparis, discovered the tots looking "like something from a horror movie" after the mauling and described it as "like a living nightmare".
Lola had facial injuries and some puncture marks on her arm, while Isabella suffered injuries to her arm, she said.
She added that both children had undergone surgery.
The twins were attacked as they slept at their parents' three-storey home at about 10pm on Saturday.