Several large companies based in Texas — including two of America's largest airliners — have said they plan to keep COVID-19 vaccine mandates for employees in place, siding with federal authorities in a showdown between the White House and the governor of Texas.

American Airlines, Southwest Airlines and IBM all said Tuesday that they would continue to require their workers to be vaccinated against COVID-19, even though Texas Gov. Greg Abbott signed an order on Monday banning private businesses from enforcing such mandates.

In a statement to The Dallas Morning News, Fort Worth-based American Airlines said that "we believe the federal vaccine mandate supersedes any conflicting state laws, and this does not change anything for American."

Dallas-based Southwest Airlines issued a similar statement to The Austin-American Statesman.

The company said they were choosing to comply with an order by the Department of Labor's Occupational Safety and Health Administration — which will require businesses with over 100 employees to ensure their workforce is fully vaccinated — because the White House order "supersedes any state mandate or law."

On Tuesday, Southwest CEO Gary Kelly told CNBC that he is opposed to vaccine mandates but that his company will still require the shots because he is required to by federal law.

"I've never been in favor of corporations imposing that kind of a mandate," Kelly said. "But the executive order from President Biden mandates that all federal employees and then all federal contractors, which covers all the major airlines, have to have a (vaccine) mandate (by Dec. 8)."

IBM told Bloomberg that it would keep in place a vaccine mandate for workers at its Texas office because the company has contracts with the federal government, meaning under Biden administration rules, it must require its employees to be vaccinated.

"IBM is a federal contractor and must comply with federal requirements, which direct employees of federal contractors to be fully vaccinated against Covid-19 by December 8th or obtain a medical or religious accommodation," a spokesperson for the company said. "We will continue to protect the health and safety of IBM employees and clients, and we will continue to follow federal requirements."

The resistance from high-profile employers could set up a legal battle, should the state of Texas choose to file a lawsuit.

According to The Associated Press, new cases of COVID-19 and hospitalizations linked to the virus have fallen in recent weeks. However, the state has seen more than 67,000 total coronavirus deaths since the start of the pandemic.