SEATTLE—Boeing Co. and its largest union have entered negotiations over where the planned 777X jetliner will be built, a decision that pits the aerospace giant's traditional manufacturing base in Washington state against its recently opened nonunion facility in South Carolina and other sites around the U.S.

Boeing on Monday confirmed the negotiations with the International Association of Machinists and Aerospace Workers, which represents more than 32,000 employees. The two parties had been in ongoing informal talks over plans...