SANT’AGATA BOLOGNESE, Italy—Italian super-sports car maker Lamborghini is about to test the allegiance of drivers and fans with radical changes to its product line and production methods.

Automobili Lamborghini, which since 1998 has been owned by the Audi unit of Volkswagen AG, currently makes just two sports cars: the Aventador, starting at $404,195, and the Huracan, at $241,945. With its next model, the iconic brand is attempting the delicate maneuver of broadening its market while maintaining a cachet and price tag few...