New York (CNN Business) Every plane in Southwest Airlines' fleet is a Boeing 737. But that may not be the case in the future.

CEO Gary Kelly said that the airline's board, at his suggestion, has asked its management team to study whether it should keepas its sole supplier of aircraft.

"Where we go from here is a question that we will address next year as to whether or not the strategy we have deployed for 48 years is the one that we want for the next 48 years," said Kelly in an interview on CNBC on Thursday. Kelly said Southwest will look at both the question of whether Boeing is the right partner going forward, and the wisdom of keeping only one aircraft type in its fleet.

The 737 Max, the more current version of the 737, has been grounded since March due to two fatal crashes that killed 346 people.

Southwest has 34 of the 737 Max in its fleet, the most of any airline. The airline reported Thursday the grounding has cost it $435 million through the end of September. Southwest expects the costs will continue to climb as it awaits a return to service and the delivery of additional jets it has ordered.

Reaching an agreement with Boeing over how it will compensate Southwest for damages is a key focus for the company right now, Kelly said.

Announcing that it is considering ending the single-jet policy could give Southwest some leverage in those negotiations with Boeing.

In response to Kelly's comments, Boeing issued a statement saying Southwest Airlines is a valued customer.

"We have been in constant contact with our customers to support them during this difficult time and our decades-long relationships are very important to us," said Boeing. "We will continue to work closely with them to reach a fair and reasonable outcome."

For Southwest, having two different aircraft types would raise the cost of training pilots, as well as the cost of spare parts and maintenance. Southwest's policy of having only one aircraft type has often been cited as helping it operate at a lower cost than its competitors.

Kelly said there is a safety advantage of having only the 737 in its fleet.

"Every [Southwest] pilot is an expert on the 737. So there's a lot of advantages, not just efficiency, but safety advantages to being devoted to one aircraft type," he said. "I'm not saying we're going to change. I'm simply acknowledging it is something that will need to be reviewed."

Southwest looked at its single jet policy at the time that Boeing announced plans to launch the 737 Max in 2011, Kelly told investors Thursday. But this is the first time Southwest has announced a board-approved reconsideration of its single jet policy.

"I feel we are obligated to do is just debate the wisdom strategically of having a sole source vendor," he said in a press call after speaking with investors.

"We may discover, I doubt it, but we may discover that it is better to have two airplanes economically and operationally," he said.

But Southwest will be forced to consider a change at some point in the future if Boeing decides to end production of the 737 Max, according to aerospace analyst Richard Aboulafia of the Teal Group.

"The 737 Max will certainly be the last 737," said Aboulafia. "Thus within the next 10 years, they'll need to prepare for a future with a completely different type, for the first time in the airline's existence."

Kelly said the review won't take place until it is able to get the 737 Max back in service.

He still believes that Boeing is a great company. "But I've been very clear, we're not happy about our situation," he said.