An apparatus for mounting a tire on a wheel rim includes a mounting head with a tire bead deflector and a tire press-in roller, and a drive that moves the mounting head toward, and rotates the bead deflector and the press-in roller around, the wheel rim. To avoid a damage-causing contact between the bead deflector and the wheel rim, an arrangement for monitoring a minimum spacing distance between the wheel rim and the mounting head is arranged preferably on the bead deflector. If an improper working position closer than the minimum spacing distance is detected, the arrangement produces a signal that influences, e.g. stops, the motion of the mounting head. The arrangement preferably includes a contact member of a comparatively soft low-friction material at a working end of the bead deflector. If the contact member contacts the wheel rim, this actuates a piezoelectric element to generate the required electrical signal.