The airline's board also intends to appoint four new directors, which would potentially include candidates proposed by Elliott Investment Management in near term.
Elliott has been pushing to revamp the airline's board and oust top executives to help improve the carrier's performance. Jordan said in June Southwest has a "great plan", which management would execute, and that he had no plans to resign.
Southwest Airlines has struggled to remain profitable since the COVID-19 pandemic and taken measures to facilitate a turnaround, including adding seats with more leg room and dropping its marquee open seating system to move to assigned seats.
It has also been hit hard by Boeing's (BA.N) jet delivery delays and is reeling from elevated operating expenses, including high labor and aircraft maintenance costs.
Six current directors on Southwest's board also conveyed their intention to step down after the airline's scheduled board meeting in November.
Last month, Elliott nominated 10 director candidates to Southwest's 15-person board including former Virgin America CEO David Cush and Robert Milton, the former CEO of Air Canada (AC.TO).
The airline said on Tuesday it intends to continue its constructive engagement with Elliott toward a collaborative resolution in the near term.
Elliott did not immediately respond to a Reuters request for comment.
