Drivers for Uber and Lyft are asking a judge to order the ride-hailing companies to pay them sick leave — as employees, not contractors — to protect both their customers and the public from possible exposure to the coronavirus.

Without paid leave, drivers “will need to work while sick to make ends meet,” a lawyer for Lyft drivers in California said in a filing late Tuesday in San Francisco Superior Court.

“I can’t self-quarantine because not working is not an option,” an unnamed Lyft driver was quoted as saying in a court declaration. “If I don’t make enough money, I can’t feed my children for the next six weeks. I’m not stopping, fever or no fever.”

After Uber drivers made a similar filing, Uber moved to transfer the case from the state Superior Court to federal court. Attorney Shannon Liss-Riordan said Wednesday she would ask a federal judge for a prompt hearing on an injunction that would require the companies to treat the drivers as employees for the purposes of sick-leave benefits.

This is “an extreme case” that requires immediate court action, Liss-Riordan wrote. “In light of the COVID-19 outbreak and the state of emergency” declared in both the state and the nation, she said, the companies’ conduct “poses a significant, imminent risk to ... employees and the general public.”

In response, Uber and Lyft both noted that they have agreed to sick-leave pay for drivers who have been diagnosed with the virus or placed under quarantine. Uber also makes payments to drivers who have been removed from work because of medical symptoms, or because public health officials said they had been exposed to someone with symptoms of the virus.

Julie Wood, a spokeswoman for Lyft, also said that “the vast majority of people who earn by driving on our (service) use Lyft to supplement existing income” from other jobs.

But Liss-Riordan said the benefits paid by both companies are less generous, and less reliable, than sick leave that state law requires for employees. In her court filing, she asserted that the company “should not get to decide what level of compliance with workplace protections it deems sufficient.”

For most workers, California requires employers to providethree days of paid sick leave per year for full-time employees. Employees can gain credit for up to six days of paid leave in a year, but can be limited to three days off with pay in a given year.

Uber and Lyft classify their thousands of drivers as independent contractors who are ineligible for the benefits state law mandates for employees, including minimum wages, overtime, pad leave and reimbursement for work expenses, such as fuel costs.

AB5, a California law that took effect this year, classifies workers as employees if they are in the same business as the companies they work for. Uber and Lyft contend their business is not transportation, but merely connecting drivers and passengers, an assessment rejected by several federal judges — most recently U.S. District Judge Richard Seeborg of San Francisco, who said in an Uber case last Friday that the company’s description “strains credulity, given the company advertises itself as a ‘transportation system.’”

But no judge has yet decided whether AB5 applies to Uber and Lyft. The companies are investing millions in a proposed November ballot initiative that would exempt them from the law.

Both companies have also contended their drivers’ contracts require any workplace disputes to be resolved by arbitrators, not judges. In Tuesday’s filing, Liss-Riordan contended the public-interest injunction she seeks is legally exempt from arbitration, and that the federal law mandating arbitration in such cases does not apply to transportation workers.

Correction: When first published, this article misstated California’s limits on paid sick leave. State law allows employers in most cases to limit employees to three days of paid sick leave per year, and allows employees to accrue up to six days that they can use in future years.

Bob Egelko is a San Francisco Chronicle staff writer. Email: begelko@sfchronicle.com Twitter: @BobEgelko