In a show of openness, the office which administers the site, the Centre for National Monuments, has opened an online poll allowing people to vote for who should be next. The results will be handed to President Francois Hollande at the end of the month.
"It's a way of involving a large part of the population - using the internet - in a choice that concerns everyone," says the president of the Centre for National Monuments, Philippe Belaval.
The survey has become a battleground for lobby groups pushing rival candidates, with the loudest campaign being run by feminists who say it is time to right a historic wrong.
Only two people in the Pantheon are women, and only one, the double-Nobel-winning physicist and chemist Marie Curie, is honoured in her own right.
The other is Sophie Berthelot, a scientist who was interred with her famous chemist-turned-politician husband, Marcellin, in 1907. She was the first woman to be pantheonised, but only "in tribute to her conjugal virtue", as the couple had been inseparable for 46 years and she died within a few hours of her husband.
"French history has tossed aside recognition of thousands of women who, by their life of service to science, the arts, philosophy, politics, or through their political or campaigning commitment, have written the story of our country and helped the advance of humanity," says the Collective for Women in the Pantheon.
"It is only justice to pay tribute to them."
Those being championed include Olympe de Gouges, an 18th-century campaigner for social reform who also demanded divorce rights for women, Louise Michel, a legendary anarchist and heroine of the 1871 Paris Commune uprising who has a metro station in her name; and Simone de Beauvoir, existentialist philosopher and author of the feminist work The Second Sex.
"The Pantheon was originally built as a church to honour Genevieve, the patron saint of Paris. That would certainly argue for letting a few more women in," Thora van Male, a lecturer at the Institute of Political Studies in Grenoble, told the Herald.
"If they don't honour women of our time, they may do so for someone from the past ..."
Pantheonisation, because it involves a choice by a single person, reflects the political moods and intellectual fashions of the time.
As a result, some of the names on the tombs are quite obscure today. Even if the honour is supposed to be eternal, some residents have been turfed out when their reputations are changed by time, such as Jean-Paul Marat, a bloodthirsty radical during the time of La Terreur.
Feminists have high hopes that Hollande, a socialist, will do the right thing. But there is no guarantee; the results of the survey are non-binding, and some campaigners suspect murky dealings.
This year Hollande mused that the time was right for a gender reboot at the Pantheon ... but then, a year ago, he favoured granting the philosopher Denis Diderot, a star of the Age of Enlightenment who is being backed by several bigwigs in the Socialist Party.
An about-turn would be difficult, especially as this year is the bicentenary of Diderot's birth and granting him a place in the Pantheon alongside Voltaire and Rousseau - the other heroes of his period - would be a fitting celebration.
Above the pillared entrance to the Pantheon, the inscription reads in French: "To its great men, a grateful fatherland." But to great women? C'est la bonne question ...
