The Future Circular Collider (FCC) design study is aimed at assessing the physics potential and the technical feasibility of a new collider with centre-of-mass energies, in the hadron-hadron collision mode including proton and nucleus beams, more than seven-times larger than the nominal LHC energies. An electron-positron collider in the same tunnel is also considered as an intermediate step, which would provide the electron-hadron option in the long term. First ideas on the physics opportunities with heavy ions at the FCC are presented, covering the physics of Quark-Gluon Plasma, gluon saturation, photon-induced collisions, as well as connections with ultra-high-energy cosmic rays.