Activist investor Elliott Investment Management and Southwest Airlines are close to a settlement deal that would give Elliott a sizable representation on the company's board, according to a person familiar with the matter.

The settlement could be announced as soon as Thursday, said the person. Spokespeople for Elliott and Southwest declined to comment.

Elliott and Southwest as recently as last week had been girding up for a proxy fight. The activist had been seeking to install 10 new directors to the carrier's board and had called for a special meeting in December to elect them. Elliott's campaign hinged in large part on the removal of Executive Chairman Gary Kelly and CEO Bob Jordan.

Elliott had said Kelly and Jordan, both three-decade Southwest veterans, should be replaced. In September, Southwest said Kelly would step down next spring, but the airline's board has staunchly backed Jordan.

CNBC could not learn whether Jordan would remain as chief executive under the terms of the potential deal, which was reported earlier by Bloomberg.

The two sides have gone back and forth with potential settlement offers over the last few months, Southwest has previously said, but until Saturday, when news broke that a settlement deal was in the offing, there was no indication that a deal was possible.

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

— CNBC's Leslie Josephs contributed to this report.