In future hadron colliders with collision energies greater than the energy of the Large Hadron Collider (LHC), the beamscreen becomes an increasingly important source of beam coupling impedance. This may lead to coherent beam instabilities, especially in the transverse plane, and potentially a failure to reach the desired beam intensity. Here we discuss design choices affecting the beamscreen impedance in the proposed Future Circular Collider (FCC-hh). We consider the resistive impedance of the copper walls and their high-temperature superconductor alternative, the impedance due to the surface treatment for electron cloud suppression, and the geometric impedance of the pumping holes and the interconnects.