By Tim Hepher and Kyle Peterson

PARIS/CHICAGO (Reuters) - A top-level exchange between Southwest Airlines Co and Airbus on the commercial success of a new aircraft has raised hopes at the European plane-maker that it might someday break Boeing Co's monopoly at the low-cost U.S. carrier, sources close to the matter said.

Southwest <LUV.N>, the world's largest low-cost airline, which celebrated its 40th anniversary in June, has for decades flown only Boeing <BA.N> 737s and has usually shut the door to EADS unit Airbus <EAD.PA>.

But after Airbus swept the board at the Paris air show in June with a revamped A320 airliner, Southwest Chief Executive Gary Kelly congratulated Airbus CEO Tom Enders in what some see as a discreet but carefully calibrated overture.

"We are exchanging friendly correspondence, which is a new dimension," a source close to Airbus told Reuters.

Boeing is seen as likely to go all-out to keep Southwest, its biggest customer, and most analysts say Airbus has only a slim chance of breaking its U.S. rival's hold.

A second source stressed that while highlighting the success of Airbus's new A320neo, the letters did not call for negotiations or do much more for now than break the ice.

But the timing of the step has attracted attention as the industry scrutinizes Southwest's actions while Boeing and Airbus offer upgraded fuel-saving models with new engines.

The sources declined to be identified because the letters exchanged between the two CEOs are confidential.

A sale to Southwest, which has long preferred the simplicity of flying just the Boeing 737, would be an even bigger coup.

Airbus and Southwest declined to comment.

"We simply don't speak publicly about potential or future business dealings," said Brad Hawkins, a Southwest spokesman.

The Southwest-Airbus correspondence came after the air show at a time when Boeing was in flux over product strategy but before the pivotal American Airlines deal which forced Boeing to respond with a plan to outfit the 737 with a more fuel-efficient engine, the sources said.

Kelly on Thursday welcomed Boeing's decision to revamp the 737.

"I applaud Boeing for that decision," he told CNBC television. "We're the world leader in the 737, so obviously it's in our selfish interest to see that aircraft improve."

In an interview with Fox Business Network, Kelly said Southwest was not in talks aimed at placing an order with the A320neo. "We're talking exclusively with Boeing and that's only fair because that's what we want to make work," he said.

OTHER OPTIONS?

Southwest on Thursday announced a weaker-than-expected quarterly profit, and its shares tumbled over 7 percent in a broadly lower market.

On a conference call with analysts to discuss the earnings, Kelly said the airline knew about Boeing's re-engining decision in advance -- "which we appreciate."

"We are anxious to sit down with Boeing and understand just what this re-engined airplane is all about," Kelly said. "Boeing continues to be our primary focus. If Boeing can meet our needs, then we can make a commitment. If they don't, well, it's a free country, and we will consider what other options we have."

Kelly said earlier this year the carrier could one day add different types of plane to its fleet.

Canada's Bombardier Inc <BBDb.TO> said in June that it had made a pitch to Southwest for its CSeries planes.

"You get better deals if people know that you have options. So talking to Boeing's competitors, of course, makes perfect financial sense," said Adam Pilarski, senior vice president at AVITAS, an airline consulting company.

"You would have to be nuts not to," he said. "It doesn't mean you necessarily buy. If they buy, it is big news."

Alongside Southwest, Delta Air Lines Inc <DAL.N> and United Continental Holdings Inc <UAL.N> also need to replace older planes.

Henri Courpron, CEO of American International Group Inc <AIG.N> unit International Lease Finance Corp, which leases planes to airlines, said Airbus has shown it could make inroads into Boeing markets. But he doubted the company would severely erode Boeing's footing.

"Ultimately, I expect the re-engined 737 and the Airbus neo to enjoy the kind of market share split that they've had over the years," Courpron told Reuters. "There will be defectors from both camps. But I don't expect that any of the products will run away with a significant market share advantage."

(Additional reporting by Karen Jacobs; Editing by Alexander Smith, David Cowell and Gerald E. McCormick)