After more than four months of waging a public battle with activist investor Elliott Investment Management, low-cost pioneer Southwest Airlines. may

Elliott first disclosed a roughly $2 billion stake in Southwest in June. Since then, it has increasingly called for changes at the airline and blamed leadership for a stagnating share price. As of Oct. 18, Elliott reported having a "combined economic exposure" to 11% of Southwest's common stock.

This could end a months-long struggle between the Dallas-based airline and the West Palm Beach-based hedge fund.

Elliott Investment Management and Southwest Airlines have begun discussing a potential settlement that would avoid a proxy fight for control of the airline's board, Bloomberg News reported on Saturday,