To expand on tdracer's excellent post, at the time the 777-200X and 777-300X were being developed, all three engine manufacturers were asked for proposals for a more powerful engine.
GE had spent some USD 2.5 billion on the GE90, mostly in the belief that only RR could deliver an engine (they believed PW could not grow the PW4000 family to the necessary thrusts - PW actually delivering the PW4090 came as a rude shock), but it was trailing RR and Pratt in sales due to technical issues (fuel injector life, for example, was horrendous. Some customers had to replace at least one injector after every flight). Pratt, meanwhile, had fumbled badly with the PW4098, which missed it's fuel burn targets by 4% and the fuel-injector swirl issue on the PW4090 resulted in very high NOx levels, which impacted how much thrust could be generated (the PW4098 received the TALON 'low NOx' combustor to try and address this).
Throw in the expectation that the 777-200X and 777-300X would only see modest sales and only Rolls-Royce was willing to accept multiple engine vendors on the 777-200X and 777-300X program. GE and Pratt both demanded exclusivity, so no matter who won, the 777-200LR and 777-300ER would have had only one engine.
Of the RR customers, only CX and AA said they'd not buy a 777X powered with a GE engine and while AA held out longer than CX...
So McNerney went to GE's board and made the proposal to effectively buy their way on the 777X program as an exclusive supplier.
GE's board approved and the rest, as they say, is history.
