The campaign Let’s waste no more time was launched on Friday by former Health Minister Philippe Douste-Blazy and infectious diseases specialist Prof Christian Perronne, to allow wider access to the drug.
According to official guidelines it may currently only be used in hospital for patients who are already at an advanced stage of illness.
The petition draws on scientific data plus decisions by both the Italian and US governments to support a call for the government to grant greater access to the drug.
It also states that death rates among hospitalised patients are low in the Marseille area, where Prof Didier Raoult has championed use of the drug at an early stage.
In a YouTube video Mr Douste-Blazy, who was France's health minister in 2004-2005, calls on people to sign the petition and explains the benefits of the medication.
"This is not about whipping up a political scandal, but rather an appeal to let hospital doctors decide to follow their consciences and give it to patients," Mr Douste-Blazy told Le Parisien.
The cause is also backed up by an online survey published by Le Parisien yesterday, which showed that 59% of the 1,016 respondents believe the treatment is ‘effective’ against coronavirus.
However, 55% believe the drug should be reserved for the most serious patients and not just those showing mild symptoms.
At present, the results of a European clinical trial called ‘Discovery’ are expected by the weekend on four treatments one including hydroxychloroquine.
Health Minister Olivier Véran has urged caution. He stated that the first results of clinical studies will determine whether chloroquine, as well as other potential treatments, will be effective when they are taken at the start of the disease.
Medicines Agency ANSM warned last Monday that treatments tested against Covid-19 could cause serious adverse effects and should "in no case" be used for self-medication.