Their spouses or family members who pressure them into wearing the covering would also face penalties.
The lower house is considering outlawing the wearing of a burqa or the niqab even though few women in France wear such body or head covering.
France is the latest of several European countries, including Belgium in April, that have moved to ban the burqa. Belgium's lower house unanimously voted to ban it in public places but the bill must now pass the upper house.
Such a ban is controversial with the public but is backed wholeheartedly President Nicolas Sarkozy who has said they are "not welcome" in France.
"In our country, we cannot accept that women be prisoners behind a screen, cut off from all social life, deprived of all identity," he said in June 2009, in the first presidential address in 136 years to a joint session of France's two houses of parliament.
"The burqa is not a religious sign, it's a sign of subservience, a sign of debasement, I want to say it solemnly," he said.
France's prime minister also favors a ban. Last week Prime Minister Francois Fillon said Muslims who wear the full veil were "hijacking Islam" and create a "dark sectarian image" of the religion.
In January this year a special commission set up to examine the wearing of the burqa reported that access to public services and public transport should be barred to those wearing the veil.
If legislation were to be passed, it would be the second time veils have been forbidden in France. A law in 2004 against the wearing of clearly visible religious symbols in schools effectively banned students from donning the burqa.
There is some confusion as to what exactly would be banned and even among Muslims what constitutes a burqa isn't always agreed.
A burqa generally refers to the long black cloak covering head and down to the ground. It often includes a niqab, the short piece of material covering a woman's mouth and nose, leaving only eyes exposed.
The less controversial hi-jab is a headscarf that covers the head but leaves the face completely visible.
Security issues will sway some people to vote for a ban. In 2006 man wearing a burqa robbed a jewelry store in the British provincial town of Bury.
But France's smaller political parties including the Green Party and Communist Party, as well as pressure groups, have cautioned against a ban, saying that it would stigmatize Muslims.
The Socialist Party would accept a partial ban in public offices and shops only.
Amnesty International said a ban would "violate the rights to freedom of expression and religion."
France's lower house will vote on the proposed bill next week but it then must be passed by the upper house, possibly in September.
The debate comes as Marseille, long a magnet for Islamic immigrants from Africa and the Middle East, is set to have France's largest mosque.
Awaiting planning permission is the so-called Cathedral Mosque, a proposed $60 million building with 80-foot minarets and prayer rooms capable of holding 7,000 people.
One councilor said it is welcome because it means that people will now be able to pray proudly, in the open and not "forced to pray in their cellars."
Marseille has only a handful of mosques even though there are an estimated 60 places of worship for Muslims.
Makhete Cisse, of the Association of Mosques in Marseille, said he would welcome the mosque, even though it would be built on the site of a derelict abattoir for pigs. The animal is deemed unclean by Muslims who are banned from eating pork products.
Stephane Ravier, of the right wing Front National in Marseille, said it isn't welcome, adding Muslims in France should respect the Christian heritage in France just as Christians should respect the Muslim heritage in a Muslim country.