The U.S. plane-maker would offer a modified version of the larger and more powerful LEAP-1A engine used on Airbus's A321neo rather than the LEAP-1B used on the 737 MAX 9, they said.
That would enable Boeing to add range while lengthening the 178-seat jet to fit 12 or more extra passengers and gain a capacity advantage over the 185-seat A321neo, the sources said.
Boeing disputes its rival's claims about the strength of demand in this niche where Airbus has the strongest lead. But leapfrogging Airbus's A321neo offering with more seats would hedge Boeing's position as many airlines opt for bigger planes.
However, the new derivative version of the 737 MAX 9, nicknamed 737 MAX 10, would bring significant headaches.
Adding the larger engine would mean raising and possibly repositioning the landing gear and recertifying parts, costing an estimated $1-2 billion, according to industry experts.
Boeing's 737 MAX family uses the smaller LEAP engine partly because its fuselage sits lower to the ground and must therefore have a smaller engine fan.
Having a different engine on the largest 737 could weaken the advantage of commonality with the smaller LEAP used on the rest of the 737 MAX fleet, but reflects a growing pragmatism in the face of lost sales.
"It doesn't matter if they are not consistent," said Adam Pilarski, senior vice president at U.S. consultancy Avitas. "They are getting killed."
The maker of the LEAP engines, CFM, which is co-owned by General Electric and France's Safran, declined to comment. A GE spokesman said there was no contractual impediment to using a larger engine for Boeing.
"The LEAP engine was designed to have growth capability," he said.
Boeing Commercial Airplanes CEO Ray Conner said this month Boeing was studying several options for developments and had made no decisions, but would support value for shareholders.
Recent orders by Vietnam start-up airline VietJet illustrate the Airbus-Boeing fight for narrow-body sales. At last November's Dubai Airshow, Airbus celebrated the sale of 30 A321s to VietJet, while Boeing officials watched from the sidelines.
But last week Boeing pulled off what industry observers saw as a coup by signing an $11 billion order for 100 737 MAXs with the same airline in the presence of President Barack Obama.
Boeing's plans to offer a larger MAX are one option being considered to defend its 737 franchise as it also tries to carve out a space in the middle of the market between the workhorse narrow-body 737 and big wide-body jets like the 787 Dreamliner.
Stung by Airbus's gains in orders, Boeing is pondering a mixed bag of tactical and strategic moves that, if fully made, could give it a head start in the development of the next generation of jets for production from about 2030.
Narrow-body medium-haul jets like the 737 and A320 family dominate the market by volume, with Airbus forecasting 22,900 deliveries worth $2.2 trillion over the next 20 years.
While both have sold thousands of the jets to airlines eager to cut fuel costs, Boeing's share of the market for such jets has fallen to 40 percent compared with a usual 50-50 split.
Market sources say Boeing has shown increasing willingness to compete aggressively for Airbus customers in order to claw back market share, as was evident in the VietJet deal.
"We expect to see lower pricing from Boeing on the MAX," Stuart Hatcher of valuation firm IBA told a briefing.
Boeing's tactical tinkering with the 737 also includes tweaking a smaller model to suit two key buyers. And the company's aim extends to a strategic 'middle of the market' jet, partly to replace its popular 757.
Industry sources say that project involves not one jet but two. They would have twin aisles and carry 220 and 260 people respectively, equating to what analysts see as two distinct slices of potential demand.
The smaller base model would have a range of about 4,500 nautical miles, dropping to about 3,500 for the larger variant.
Airbus has dismissed the idea, which would partially overlap with its A321neo. It is expected to tell media at briefings this week that the history of the market is littered with small twin-aisle jets that sold poorly, including its own A310.
But it is holding in reserve a plan to retaliate with another A321 makeover, using new wings to boost performance.
Boeing declined comment on either tactical plans to defend the 737 or the longer-term mid-market studies. At $15 billion or more, it has said the latter is a difficult business case.
"We're in continuous discussions with our customers about the market. We'll make the right decisions at the right time," a spokesman said.
While Boeing's mid-market study is grabbing most industry attention, behind it lies a longer-term bid to turn the tables on Airbus in the broader market for smaller jets where both make most cash, according to industry analysts.
Although differing in size and appearance, a mid-market jet would spawn new technology and production methods that could be transferred to whatever comes after the 737, they said.
(Tim Hepher and Alwyn Scott - Reuters / Yahoo Business News)