Southwest and Elliott are close to a settlement that would end the proxy fight, sources say

Activist investors Elliott Investment Management and Southwest Airlines A settlement agreement that would give Elliott a larger representation on the company's board is close, according to a person familiar with the matter.

The settlement could be announced by Thursday, the person said. Spokesmen for Elliott and Southwest declined to comment.

Elliott and Southwest were poised for a proxy fight as recently as last week. The activist wanted 10 new directors on the carrier's board and called a special meeting in December to elect them. Elliott's campaign was largely about ousting executive chairman Gary Kelly and CEO Bob Jordan.

Elliott 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 strongly supported Jordan.

CNBC has not learned whether Jordan will remain as chief executive under the terms of the potential deal, which was previously reported by Bloomberg.

The two sides have bounced back and forth over potential settlement offers over the past few months, Southwest has previously said, but until Saturday, when news broke that a settlement agreement was off, there had been no indication that a deal was possible.

Southwest reports third-quarter results before markets open Thursday. Jordan and other executives will hold a call with analysts at 12:30 pm ET.

— CNBC's Leslie Josephs contributed to this report.