The disclosed method produces a spline-grooved cylindrical member, e.g. torque converter front cover, including a cylindrical portion having a target wall thickness t2, from a plate blank having a plate thickness t1. An outer peripheral portion of a disc-shaped blank is plastically deformed by pressing it against a mandrel with the concave peripheral edge of a thickness-increasing roller, so that a thick wall portion, of a thickness greater than the plate thickness t1 of the blank, is formed in the cylindrical portion. The thick wall portion is then pressed against the mandrel by forming rollers to form spline grooves therein. Thus, the other portions of the product have a wall thickness which is not unnecessarily increased. In this manner, the wall thickness t2 of the cylindrical portion is not entirely dictated by the wall thickness t1 of the blank material. Thus, a cylindrical member resistant to high pressures acting on its axially-extending outer cylindrical portion can be produced with an optimal configuration.