With the share of firm orders for re-engined single-aisle aircraft still tilted heavily in Airbus's favour, Boeing has come to the Paris air show intent on showcasing the slow-selling 737 Max 9 and launching the largest version of the 737 yet, the Max 10.

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Less than two months from its debut flight, the first 737 Max 9 flew across the Atlantic to appear in the static display at Le Bourget. Not far away in the static park, Airbus is displaying its A321neo, the aircraft that provoked Boeing to launch a new and even larger version of the 50-year-old single-aisle with the proposed 737 Max 10X.

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Boeing's predicament in the single-aisle orders battle is clear. As of the end of May, Airbus has amassed a firm order backlog of 5,054 A320neo-family aircraft compared with Boeing's tally of 3,714 for the 737 Max family.

Three different people have led Boeing Commercial Airplanes since the 737 Max was unveiled nearly six years ago, but the 737 Max's slice of the single-aisle market relative to the A320neo has remained stubbornly at or below 40%.

The disadvantage is felt by the 737 Max 8, but is most acute at the larger end of the single-aisle range. The A321neo's 20-seat advantage over the 737 Max 9 in single-class configuration has led to an embarrassing imbalance in market share, with the Airbus product enjoying a six-to-one sales lead.

In response, Boeing acknowledges the deficit but quibbles with the details, questioning whether a portion of orders in the Airbus backlog will ever deliver to airlines with potentially unsustainable growth rates.

Returning the single-aisle backlog to the traditional parity between the A320 and 737 is, anyway, not a priority for Kevin McAlister, the newly appointed president and chief executive of Boeing Commercial Airplanes. McAllister's sales chief, Ihssane Mounir, may attempt to win every deal, but not at any price.

"I wake up focussed on making sure as a business we do the right thing. That means when we win in the market, we win with scaleable growth. We win the right spots," McAllister says. "So I don't sit back and say [we must achieve single-aisle order] parity by a certain date. That's the right long-range strategy in the single-aisle market."

Although strict parity is not a goal, Boeing last year felt obliged to make some effort to recover from the A321neo's dominance over the 737 Max 9. Mike Delaney, Boeing's vice-president and general manager for aircraft development, initially told his team of engineers to design a longer version of the 737 that could accommodate 240 seats in a standard single-class layout, like the A321neo.

Technically, it was a challenging assignment. The 737 is designed to sit lower to the ground than the A320. That reduces the length and weight of the landing gear and makes it easier for baggage handlers to gain access to the baggage compartment. But that feature also makes it harder to lengthen the 737 fuselage or enlarge the engines hanging under the wings without the risk of scraping the tail or the bottom of the engine nacelles on the runway as the aircraft rotates during take-off.

In response, Delaney's team came up with a new concept. Featuring larger CFM International Leap-1A or Leap-1C engines and special telescoping landing gear, the "737 Max 10X" concept Delaney showed to customers last summer carried as many passengers in single-class as the A321neo.

As the 737 fuselage is 25.4cm (10in) narrower than the A320 family, Boeing's concept was slightly lighter and more aerodynamic than the A321neo, Delaney says, but the difference was small.

Within months, however, Boeing had withdrawn the concept, but not, according to Delaney, because of the cost or complexity of the development effort. Rather, Boeing's customers expected the 737 Max 10X to be significantly lighter and more efficient than the Airbus competitor.

"I told them, I have a really great engineering team but they still have to live within the laws of physics of the universe," Delaney says. Compared with the A321neo, the initial 737 Max 10X had the same wingspan and the same engine, so the weight of each aircraft was similar. With similar weights and wingspans, plus identical engines and seats, there was no way for Boeing’s original concept for the 737 Max 10X to offer significantly better performance than the A321neo.

"How did you think I was going to get an airplane down there?" Delaney says he asked his customers.

As the discussion continued, Delaney recalls a breakthrough realisation. Boeing had assumed that customers most of all wanted the same number of seats as the A321neo in a single-class layout. Instead, what they really were asking for was a new aircraft with lower costs on a seat-mile basis, even if that aircraft offered 10 fewer seats in a standard single-class layout.

“We said: 'That was a different problem,'" Delaney recalls.

Delaney's team went back to the drawing board. By October last year, a new concept for the 737 Max 10X had emerged. It was only about 1.7m (5.5ft) longer than the 737 Max 9. That was enough to carry 230 seats in a single-class layout and 188 in a dual-class cabin. But it didn't need bigger and heavier engines, allowing Boeing to claim a 5% lower seat-mile cost than a standard-range A321neo.

The additional length requires Boeing to integrate a semi-levered landing gear similar to the design already used on the 777-300ER and 787-10. Additional tests on the new landing gear design are expected to be completed by the end of the year.

Boeing's 737 Max 10X concept still doesn't match the capacity of the fast-selling A321neo, but that was by design. Instead of competing with Airbus on single-class seating capacity, Boeing is offering a slightly smaller aircraft and claiming better efficiency.

"This is not a 'me too' airplane versus the A321," he says. "I think it creates parity on seats, which has been a hole in the Max family. But our customers wanted us to take not only the 737 Max 9 and make it bigger, but they want it better."

The Paris air show provides an ideal venue to find out how much interest the new 737 Max 10X concept has actually generated.

Speaking to journalists in early June, Mounir – the vice-president of global sales and marketing – said the additional seating capacity and fuel efficiency of the new concept resonated among potential customers, including some that have already bought the 737 Max 9.

"I'm talking to more than a dozen customers right now on the Max 10," Mounir says. "Several people are considering having both the 9 and 10."

Neither the 737 Max 9 or 737 Max 10X concept is aimed at the A321LR, the minor variant equipped with three auxiliary fuel tanks to fly routes now operated by 757-200s up to 4,000nm (7,400km). Instead, Boeing is positioning both aircraft to compete against the standard A321neo.