Pakistan's Supreme Court accepted Bibi's appeal against a conviction for blasphemy last week but, after thousands of Islamists demonstrated and rioted against the ruling, the government agreed to start the procedure to ban her from leaving the country.
Her family is in hiding and her lawyer, Saiful Mulook, has left the country, saying his life was in danger.
An alliance of Islamist parties has declared it will stage a "million-man march" to protest at the acquittal in the port city of Karachi on Thursday.
"France has a duty to come to her assistance and her family's in intervening with Pakistan so that it authorises her to take refuge in another country where she will be safe and where she can practise her Christian religion freely," Hidalgo told the AFP news agency on Tuesday.
The Paris city council made Bibi a citizen of honour in 2014 "to contribute to her protection" and Hidalgo said she was ready to provide refuge for her and her family, "so that they can settle in France for good".
Call for asylum
"Today a woman needs France" Laurent Wauquiez calls for help for Asia Bibi
In a video posted on social media, Wauquiez, the leader of the Republicans party, called on President Emmanuel Macron to "do everything necessary to save [Bibi's] life" and "make it possible for her to come to France and be protected".
"This is about a woman's life but this is also about our idea of civilisation in the face of Islamist barbarism," he declares in the message.
Wauquiez, who takes a tough line on immigration, tells the president that "it is to France's credit that it has always offered protection to the victims of oppression at moments like this".
"At the same time we must ensure that the right of asylum is not abused by those who don't have the right to it. France must also offer generosity to those who are the real victims of Islamist babarism," he goes on.
"Her life is in danger," Wauquiez says. "France cannot remain silent."