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.

Activist investor Elliott Investment Management and Southwest Airlines are in talks to settle their months-long battle, according to reports Sunday, with a potential deal that would give Elliott representation on the budget airline’s board.