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.

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,

Southwest and Elliott declined to comment to Bloomberg and the publication noted the talks were not finalized and could fall through.