RENTON, WA—Boeing Co. is facing a problem as it races to meet demand for single-aisle, fuel-efficient jets: Where to store unfinished 737s piling up at its factory near Seattle.

RENTON, WA—Boeing is grappling with how to pack a third assembly line into its assembly plant here, which needs to build 737s at a record pace.

WICHITA, KS—Spirit Aerosystems is preparing to produce up to 60 fuselages a month to keep up with demand from Boeing’s 737 assembly plant in Renton, WA.