Elliott is one of Southwest's largest investors and plans to engage with the airline's management team, the report said.
Southwest and Elliott did not immediately respond to Reuters' requests for comment.
The Dallas-based airline said in April it expected higher costs and slower-than-expected revenue growth as U.S. planemaker Boeing will now be able to deliver it just 20 aircraft this year.
Elliott, one of the world's most prominent investors, is known for pushing for changes at companies to boost shareholder returns.