According to one embodiment, a paper sheet processing apparatus includes a supply unit, a pick-up mechanism, an inspection device, a stacking device, and a wrapping device configured to wind a band around a stacked paper sheet bundle and wrap the stacked paper sheet bundle. The wrapping device includes a clamp mechanism configured to curve the stacked paper sheet bundle by pressing both side portions of the stacked paper sheet bundle, a band winding device configured to wind a wrapper band around the curved stacked paper sheet bundle, and a heater configured to heat-seal an end of the wrapper band. The clamp mechanism includes a press member configured to adjust a degree of curvature of the stacked paper sheet bundle.