Pulley device for a rotary machine, and in particular a compressor, comprising a pulley, a rolling bearing, a torque transmission plate having an axial sleeve designed to be mounted on a rotary spindle of the rotary machine and a radial plate extending radially from the axial sleeve and having the general shape of a disc, said pulley being mounted on an outer ring of the rolling bearing and on the torque transmission plate, the torque transmission plate being able to transmit a rotational movement from the pulley to the rotary spindle.Said pulley device comprises a damping element located between the pulley and the torque transmission plate.