By Tim Hepher and Kyle Peterson

PARIS/CHICAGO (Reuters) - Southwest Airlines <LUV.N> and Airbus <EAD.PA> have begun a "warm" top-level exchange, prompting the jetmaker to hope Boeing's <BA.N> biggest customer may eventually take an interest in its planes, sources close to the matter said.

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 Europe's Airbus.

But after Airbus swept the board at the Paris air show in June with a revamped A320, Southwest Chief Executive Gary Kelly congratulated Airbus counterpart Tom Enders on the move 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.

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

But the timing of the step has not gone unnoticed as the industry scours the influential airline's actions while Boeing and Airbus offer upgraded fuel-saving models with new engines.

A sale to Southwest, which has long preferred the simplicity and savings of one Boeing type, would be an even bigger coup.

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

Airbus declined to comment.

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

The exchange came after the air show at a time when Boeing was in flux over product strategy but before the pivotal American deal which forced Boeing to respond, the sources said.

Kelly on Thursday welcomed Boeing's decision to revamp the 737 as the airline announced quarterly profits.

"I applaud Boeing for that decision and we're waiting to hear from them exactly what that means for Southwest in the future, so I see that only as a good thing for us. We're the world leader in the 737, so obviously it's in our selfish interest to see that aircraft improve," he told CNBC.

MOMENT TO STRIKE?

Some industry experts have raised questions over how fully Southwest was kept in the loop on Boeing's sudden decision to offer American a version of its 737 featuring a new fuel-efficient engine to avoid getting locked out of the deal.

The surprise move won Boeing an order for 200 737s from American and split the $40 billion order with Airbus, which had already committed to giving the A320 a new engine. But it may have annoyed Southwest, which built its business around the 737.

"This is definitely a moment for Airbus to strike because there may be bruised feelings at Southwest," said aerospace consultant Richard Aboulafia of Virginia-based Teal Group.

"Southwest played a key role in designing the 737 Classics and a very key role in the 737 Next Generation," he said. "Boeing's traditional way of developing airplanes is to get customers on board first, and this didn't happen."

Boeing is seen as likely to go all-out to keep Southwest.

Boeing declined to comment but said the company aims to please its "valued customer" which has bought 670 of its jets.

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

Canada's Bombardier <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."

Some sources however said contacts with Airbus had been few.

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

Henri Courpron, CEO of International Lease Finance Corp <AIG.N>, which leases planes to airlines, said Airbus had 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 expected that any of the products will run away with a significant market share advantage."