A source reports that Southwest and Elliott are close to reaching a settlement that would end their proxy fight.

• A proxy fight at Southwest Airlines is likely to be averted with a settlement between the airline and Elliott Management, according to a source.
• Elliott will receive a significant presence on the company's board, but it will not be a majority.
• Last week, there were no signs that Elliott and Southwest were planning anything other than preparing for a fight.

A source close to the matter reveals that Elliott Investment Management and are near a deal that would grant Elliott a significant presence on the company's board.

An announcement about the settlement could be made as early as Thursday, according to a source. However, representatives for Elliott and Southwest refused to provide any comment on the matter.

Recently, Elliott and Southwest had been preparing for a proxy fight, with the activist seeking to install 10 new directors and remove Executive Chairman Gary Kelly and CEO Bob Jordan from the board.

Southwest's board has staunchly backed Jordan, despite Elliott's recommendation to replace both him and Kelly, who are both three-decade Southwest veterans.

Bloomberg reported earlier that a potential deal may keep Jordan as chief executive, but CNBC could not confirm this information.

Southwest had been negotiating potential settlement offers with the other side for several months, but until Saturday, there was no indication that a deal was achievable.

Before the market opens on Thursday, Southwest will release its third-quarter results. Jordan and other executives will then hold a call with analysts at 12:30 p.m. ET.

— CNBC's Leslie Josephs contributed to this report.