In December 2010, Airbus launches the A320neo, featuring new engines (LEAP included).
In July 2011, American Airlines releases a press release  containing the following:
"As part of the Boeing agreement, American will take delivery of 100 aircraft from Boeing’s current 737NG family starting in 2013, including three 737-800 options that had been exercised as of July 1, 2011. American also intends to order 100 of Boeing’s expected new evolution of the 737NG, with a new engine that would offer even more significant fuel-efficiency gains over today’s models. American is pleased to be the first airline to commit to Boeing’s new 737 family offering, which is expected to provide a new level of economic efficiency and operational performance, pending final confirmation of the program by Boeing. This airplane would be powered by CFM International’s LEAP-X engine." (emphasis added)
In August 2011, Boeing announces the 737 MAX program, featuring LEAP engines.
American Airlines literally ordered a plane that didn't exist. And then Boeing tried to build it.
There are structural failings (e.g. why Boeing wasn't better positioned by investing in a redesign in the 2006+ period), but American shares a fair amount of blame for this clusterfuck.
Boeing certainly could have said "No", in which case American likely would have bought additional planes from Airbus. So American had leverage, they used it to pressure Boeing into building what they wanted, Boeing failed at delivering that, and we're here today.
Previous to American's press release, Boeing's leadership was strictly avoiding mentioning an up-engined 737. And was fairly reliably mentioning a clean sheet redesign program.
I can't think of a reason for doing that, if they in fact had such a program.
It would make sense to be trumpeting it loudly to anyone and everyone.