Discoseismic c modes in accretion discs have been invoked to explain low-frequency variabilities observed in black hole X-ray binaries. These modes are trapped in the innermost region of the disc and have frequencies much lower than the rotation frequency at the disc inner radius. We show that because the trapped waves can tunnel through the evanescent barrier to the corotational wave zone, the c modes are damped due to wave absorption at the corotation resonance. We calculate the corotational damping rates of various c modes using the Wentzel-Kramers-Brillouin (WKB) approximation. The damping rate varies widely depending on the mode frequency, the black hole spin parameter and the disc sound speed, and is generally much less than 10 per cent of the mode frequency. A sufficiently strong excitation mechanism is needed to overcome this corotational damping and make the mode observable.