Incoming HP Inc. Chief Executive Enrique Lores is moving quickly to imprint changes on the computer hardware maker with plans to shrink the company’s ranks by as much as 16% in a restructuring plan that also aims to revive lagging printer sales.

HP Thursday said it could eliminate 7,000 to 9,000 jobs from its roughly 55,000 workforce over the next three years. The cuts, once completed, should yield annual savings of about $1 billion, the company said at its annual securities-analyst meeting. HP is nearing the end of a three-year-old...