"They asked for 48 hours to put together plans to safeguard workers in their facilities," Gamble said, adding that the window expires this afternoon.

In a letter to union members, UAW President Rory Gamble said Ford , General Motors and Fiat Chrysler haven't been willing to pause production.

The United Auto Workers union has asked the Big Three Detroit automakers to shut down production for two weeks to safeguard its members from the coronavirus outbreak.

A task force meeting is scheduled for later Tuesday, and its members "have been working diligently since Sunday and that work continues," GM spokesman Jim Cain said in an emailed statement. More information will be provided after the meeting, he added.

In response to the UAW's letter, executives at Ford told CNBC that they are considering production cuts at the company's U.S.-based factories as a possible solution. The company is trying to determine if a plant would need to closed down completely in order to sufficiently reduce risk, or if it could be kept operating with reduced staffing.

One of Ford's employees at a product development office in Dearborn, Michigan, was exposed to coronavirus after leaving work several days ago, and was diagnosed before returning to work, the company said.

GM confirmed earlier on Monday that an employee at its Cole Engineering Center outside Detroit tested positive for COVID-19. The company said it is working with local officials to clean the area, and it directed some employees to self-quarantine for 14 days.

The UAW confirmed the first known confirmed case of COVID-19 in an employee of a Detroit automaker on March 12. The worker was employed at a Fiat Chrysler plant in Indiana, although a spokesperson said production at the plant would continue as normal. The company said an undisclosed number of employees who may have come into direct contact with the person were quarantined.

The company also announced Monday that it will halt operations at a majority of its plants in Europe, which the World Health Organization considers the new epicenter for the disease.

"FCA has already implemented extensive protocols to ensure the health and welfare of our workforce," Simon Sproule, a spokesman for the company, said in an email Tuesday. "We are continuing to carefully monitor the situation and are making improvements as needed."

Last week, Ford and General Motors instructed employees who can work from home to do so beginning Monday due to the coronavirus outbreak that is now spreading through the U.S.

In Canada, Fiat Chrysler halted production at its minivan plant after workers refused to work amid concerns of an employee contracting the coronavirus.

The virus has infected more than 183,000 people across the globe and has killed at least 7,100, according to Johns Hopkins University data. In the U.S, it has infected more than 5,000 and killed at least 85.

Read the UAW's letter to membership is below:

Brothers and Sisters,

I'm going to get right to the point. I want you to know exactly where we are with our discussions with the Big 3 leadership and protecting our members and their facilities during this international crisis. I announced this past Sunday that the International UAW had formed a COVID-19/Coronavirus task force with GM, Ford and FCA to implement enhanced protections for manufacturing and warehouse employees at all three companies. I want to be very transparent about what happened during our conversation Sunday with the Big 3. The UAW leadership, based on the World Health Organization (WHO) and Centers for Disease Control and Prevention (CDC) recommendations on how to protect ourselves and our communities, requested a two-week shutdown of operations to safeguard our members, our families and our communities. Your UAW leadership feels very strongly, and argued very strongly, that this is the most responsible course of action. The companies, however, were not willing to implement this request. They asked for 48 hours to put together plans to safeguard workers in their facilities. (Ford Motor Company has indicated this week that they are willing to rotate down shifts and are planning to shut down all European operations next week.) The 48-hour window is up this afternoon. We will be evaluating what the companies submit today and there will be a meeting this evening at 6 p.m., where the Task Force will review plans for the safety and health of all members, their families and our communities. I want to be very clear here: If the UAW leadership on the task force, myself and Vice Presidents Cindy Estrada, Terry Dittes and Gerald Kariem, are not satisfied that our members will be protected, we will take this conversation to the next level. These companies will be put on notice that the UAW will use any and all measures to protect our brothers and sisters who are working in their facilities. And make no mistake, we have powerful allies who have stepped up to help us. Michigan Governor Gretchen Whitmer, despite what you might have heard in some recent erroneous reports, was instrumental in assisting us in bringing the Big 3 to the table, as was U.S. Representative Debbie Dingell. I know these are very difficult days for all of us. Please know that my #1 priority, and the #1 priority of the entire UAW leadership, is the safety and well-being of our UAW family and our communities. I will be sending another update very soon on what we see from the Big 3 today. I'd like to ask that we all remember that we are all in this together. In Solidarity, Rory

— CNBC's Phil LeBeau contributed to this report.

Correction: UAW President Rory Gamble's letter to union members said Ford, General Motors and Fiat Chrysler haven't been willing to pause production. An earlier version misstated what the letter said.