Shares of the Dallas company at last check were 2.5% higher at $48.62.
Other airline stocks, such as Delta (DAL) - Get Delta Air Lines, Inc. Report, United (UAL) - Get United Airlines Holdings, Inc. Report and American Airlines (AAL) - Get American Airlines Group, Inc. Report, were also rising.
So-called Worker Adjustment and Retraining Notifications notices were issued to 6,828 Southwest employees, including 1,221 pilots and 1,500 flight attendants, the airline said in a statement, "based upon a lack of meaningful progress in negotiations."
Airlines have suffered during the coronavirus pandemic, which has severely curtailed travel due to social distancing requirements.
"Southwest Airlines has been in discussions since early October with union representatives from all of our contract groups to negotiate temporary cost reductions to help offset the billion dollars of overstaffing costs projected for next year," Southwest said in a statement.
The involuntary furloughs will take place on March 15 or April 1, "unless Southwest reaches cost-saving agreements or the government enacts a satisfactory Payroll Support Program extension."
"The workgroups that approve these temporary reductions for next year will receive protection from furloughs for all of 2021," Southwest said.
The furloughs would affect 12% of the airline's nearly 58,000 staffers.
The new warnings increase the number of people at risk of being cut between January and April 1 to 7,273, or nearly 13% of the workforce, Bloomberg reported.
“Today marks a sad milestone in the history of Southwest Airlines,” Jon Weaks, president of the Southwest Airlines Pilots Association, the labor union that represents roughly 9,000 pilots at the carrier, said in a video message.
“While this development is not completely surprising, it is incredibly disappointing to our pilots and their families.”
Weaks stressed that the notices are not furlough letters.
"Rest assured we will continue to engage with the company and we will not stop searching for a solution that represents the best interest of our entire pilot group," he said.
About 17,000 employees already have left Southwest temporarily or permanently through voluntary programs.
Southwest said it reached agreements with, and extended furlough protections for 2021 to, members of the dispatch and meteorologists workgroups.