Nine-month-olds Lola and Isabella Koupparis were attacked in their cots at around 10pm on Saturday night in Homerton, east London.
Police said the little girls were in a "serious but stable condition" at the Royal London Hospital.
They both suffered arm wounds and one is believed to have facial injuries.
Their mother Pauline said today: "We're just worried about them and making sure they're OK.
"One is really good and one is not so good."
She spoke as she left her smart three-storey terrace just after 10.30am today, carrying a packet of nappies and looking anxious.
Video: Baby girls mauled by fox
Clutching a teddy bear, she was driven away in a black BMW by the girls' aunt, Barbara Koupparis.
The aunt added: "They are OK. I really can't say anything."
The fox apparently entered through a door on the ground floor which had been left open because of the heat.
The animal attacked the twins in their upstairs room as their parents, Nick and Pauline, were reportedly watching television.
Their four-year-old brother, Max, who was also sleeping upstairs, was not hurt.
A shocked neighbour of the family, who knows them well, described the girls as "beautiful".
The woman, who lives next door but one and did not want to be named, said: "They're beautiful little babies, really beautiful.
"We've all got foxes at the bottom of our gardens. Some people have got two or three living in their garden.
"They're all as bold as brass. You walk out into the garden and you have to shoo them away."
Asked if she was worried about the safety of her own little boy, who is a similar age to the twins, she said: "Of course, it was shocking."
Michael Parra, 48, a health trainer, who has lived in the street for six years, said police had been going from house to house after the attack, warning residents not to leave their doors open.
He said: "Police told us that in the hot weather we should keep our doors closed for our safety.
"We see a lot of foxes around here. They're always in our garden.
"Something should be done about them. I would love to get them out of here. They're really a nuisance and a danger.
"They've terrorised our garden."
He said he had complained about the fox problem to the local council but nothing had changed.
He added: "I wonder how much they can do about the situation.
"I think the foxes are getting bolder. They almost go up to you.
"I've got fearful myself. They've gone towards my dog too."
After the attack pest controllers set fox traps in the back garden.
Last night a fox was discovered in one of the devices and was humanely destroyed by a vet early today.
A police spokesman said: "A vet was called to establish if it was safe to move the animal. It was determined it was not and the fox was humanely killed by the pest controller at approximately 12.15am.
"The traps will remain in situ for the time being."
In 2002 Kent woman Sue Eastwood said her baby boy, Louis, was injured after a fox crept into their house while she slept.
Louis, 14 weeks old, suffered bite marks on his head after the animal crept into the sitting room of the house in Dartford.
Neighbour Sav Remzi, 46, a music executive, said he heard one of the baby girls crying as she was carried to the ambulance after the attack.
He said: "I heard lots of noise outside on Saturday night and saw ambulances arriving and lots of police.
"I heard a child crying as they took her to the ambulance.
"I see foxes all the time and I like them generally but this is quite amazing.
"Usually I quite like the fact that they're wild animals living among us but this is really quite unexpected. I never imagined something like this happening."
Student Ozlem Pekcan, 21, who lives next door to the family, said she had discovered a fox in her house on more than one occasion.
She said: "I saw one in my kitchen once, looking for food, and I shooed it off.
"Another time there was one in my living room.
"When it was in the kitchen I looked at it and they usually run away but it was just staring back at me.
"I think they're used to going into people's gardens and meeting people. They're not really scared of you."
On both occasions the foxes had entered the house through a door that had been left open, she said.
She described the babies next door as "lovely".
Hackney Council said this was the first time it had ever had a report of a fox attacking a resident.
A spokesman said: "We usually receive around a couple of phone calls a month from people concerned about foxes getting into their bins and we always advise residents to make sure they secure their bins and their fences, and to not leave food out.
"We have never had reports in the past of foxes attacking residents. All the expert advice we have had suggests that shocking incidents like this are incredibly rare and our thoughts are with the children and their family."
The council spokesman added that rubbish collections in the area were carried out on a weekly basis.
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