Southwest Airlines CEO Gary Kelly issued a memo indicating that Southwest continues to encourage its workers to get vaccinated but does not intend to enforce a mandate. "Obviously, I am very concerned about the latest delta variant, and the effect on the health and safety of our employees and our operation, but nothing has changed," Kelly said.

While American Airlines encourages passengers and staff to receive the vaccine, CEO Doug Parker told the New York Times that a vaccine mandate was not on the company's agenda. "We certainly encourage it everywhere we can, but we're not putting mandates in place," said Parker.

Delta Airlines has also announced it would not mandate vaccination, at least not until the vaccine has been fully approved by the Food and Drug Administration (FDA). "It's very difficult for us to come in and mandate a vaccine that isn't even federally approved yet. The authorization hasn't been final yet, so stay tuned," Delta's CEO Ed Bastian told CNBC.

In a separate interview with Fox 5, Bastian explained that a significant portion of Delta Airlines staff is in fact already vaccinated.

"We're almost 75 percent vaccinated already," Bastian said. "And if you think about that, you have probably some portion – maybe call it 5 to 10 percent of our employee base – that is going to have some medical or religious reason why they're not getting vaccinated, you're really down to a relatively modest number, maybe 10 to 20 percent of the unvaccinated, that you can drive with a mandate."

In May, Delta Airlines became the first major U.S. airline to introduce a vaccine mandate for all new hires only – a decision that was also taken by United Airlines in mid-June.