NEW YORK (Reuters) - Elliott Investment Management is telling Southwest Airlines ( LUV ) investors the company needs new outside leadership and that it wants to engage to shape a "better future" but is also ready to press ahead with a planned proxy fight.

In a letter to shareholders, seen by Reuters, Elliott is dialing up the pressure on the airline by proposing a new board-level committee that would conduct a thorough business review and "drive transformational change."

The hedge fund said it is eager to meet with company representatives on Sept. 9 to discuss ways to tackle the company's "immense" challenges. But Elliott also warned that if leaders cannot identify "what is best for Southwest ( LUV )" and its stakeholders, it will push ahead with a board challenge.

Elliott has been pushing to refresh the board and oust top executives to help improve the carrier's performance.

The airline has been trying to implement a turnaround plan including adding seats with more leg room, moving to assigned seats and naming a new board member in July. Its stock price, which has been under pressure, pared losses recently.

A representative for Southwest ( LUV ) was not immediately available for comment.

Earlier this month, Elliott laid out plans to nominate 10 director candidates to Southwest's ( LUV ) 15-person board, including former Virgin America CEO David Cush and Robert Milton, the former CEO of Air Canada.

In the letter, Elliott said why it wants so many board seats. It does not want to be "in charge" but feels the board is "purpose-built to serve the interests" of CEO Robert Jordan and his predecessor and current executive chairman Gary Kelly. The hedge fund has pushed for Jordan and Kelly to be replaced since its stake in the airline became public in June.

As a public company, Elliott wrote, Southwest ( LUV ) must be accountable to shareholders and is not "an absolute monarchy."

Elliott owns roughly 8% of Southwest's ( LUV ) outstanding shares, according to a recent regulatory filing. Including derivatives, the hedge fund has an economic interest of 11% in the carrier.

Some investors share Elliott's frustration, the letter said, noting that Artisan Partners publicly called for leadership change and urged the board to work with Elliott. Other shareholders expressed concerns privately to the hedge fund, it wrote.

Elliott also pointed to frustration among Southwest ( LUV ) pilots who have called for bold leadership to fix corporate problems.

The airline must find a comprehensive solution, "not just some hand-picked new directors beholden to current management and a few long-overdue initiatives," the letter said.

While CEO Jordan recently wrote that the fight with Elliott is "a battle for the heart of our company," the hedge fund said it is a fight for Jordan and Kelly to "continue to control Southwest ( LUV ), on their terms, for as long as they wish."

Elliott has said it is preparing to call a special shareholder meeting where investors would have a chance to vote on directors.

(Reporting by Svea Herbst-Bayliss; Editing by Muralikumar Anantharaman)