Activist investor Elliott Investment Management and Southwest Airlines are reportedly nearing a settlement that would grant Elliott significant representation on the airline’s board, according to a source familiar with the situation.

An announcement regarding the settlement could come as early as Thursday, the source indicated. Representatives for both Elliott and Southwest declined to comment on the matter.

Just last week, Elliott and Southwest were preparing for a proxy battle, with Elliott aiming to add 10 new directors to the board and calling for a special meeting in December for their election. The campaign was largely focused on removing Executive Chairman Gary Kelly and CEO Bob Jordan.

Elliott argued that both Kelly and Jordan, who have been with Southwest for three decades, should be replaced. In September, Southwest announced that Kelly would step down next spring, but the board has consistently supported Jordan’s position.

It remains unclear whether Jordan will continue as CEO under the terms of the prospective deal, as previously reported by Bloomberg.

The two parties have exchanged various settlement offers in recent months. However, until news emerged on Saturday about a potential agreement, there had been little indication that a resolution was in sight.

Southwest is scheduled to release its third-quarter results before the market opens on Thursday, followed by a call with analysts at 12:30 p.m. ET featuring Jordan and other executives.

— CNBC’s Leslie Josephs contributed to this report.