French actresses Isabelle Adjani and Juliette Binoche, and NGOs such as Sea Shepherd and Greenpeace France are among those who have signed.
This kind of initiative already exists in some 40 countries around the world, but it is a first in France.
French researchers from different sectors and institutions will accompany people who sign up to the Lundi Vert online challenge by offering weekly encouragement and advice on how to change their habits.
There are three main reasons behind the launch of such a public initiative.
The first is the environment.
According to Le Monde newspaper, which published the petition on 3 January, breeding animals for meat contributes to deforestation and the loss of biodiversity.
Eighty-five percent of forests in South America have been cleared for this purpose.
The United Nations says this breeding on a large scale contributes to 14.5 percent of greenhouse gas emissions.
But not only that – there are health benefits from eating less meat in terms of limiting heart disease, cancer, diabetes and other conditions.
Also, there is the question of animal welfare.
Laurent Bègue-Shankland is a professor of social psychology at the University of Grenoble-Alpes, southeast France, who designed the questionnaire for the online campaign.
He says he hopes small efforts, day by day, not eating meat will become a long-term habit. But at first, he admits, it will be a huge cultural change for French people.
Nicolas Treich is a director of research from INRA, a national institute for agricultural studies in Toulouse. He is one of the people behind the French Green Monday initiative.
He told RFI that public policies are very limited when it comes to reducing meat consumption while there is strong climate policy addressing the transport industry, such as the carbon tax.
He says in light of the Yellow Vest protests, it's important to separate two issues: on the one hand, social problems in France and on the other, data shows there is an over consumption of meat.
Yann Arthus Bertrand is a well-known aerial photographer, whose Fondation GoodPlanet has signed the Lundi Vert petition.
He is no stranger to the Green Monday challenge. He did the photography for Meat Free Monday, a non for profit campaign set up by musician Paul McCartney in 2009.
Green Monday, he says, "is a personal contribution to the earth". It’s about values and looking out for one another, taking part in something beyond one’s personal sphere.
He says although French people are not quite ready for such a challenge, they are starting to open up to the idea of changing their habits.
And he emphasises the role the government will have to play in helping agricultural enterprises deal effectively with the transition.