Single and pair top-quark production in proton–lead (p–Pb) and lead–lead (Pb–Pb) collisions at the CERN Large Hadron Collider (LHC) and Future Circular Collider (FCC) energies, are studied with next-toleading-order perturbative QCD calculations including nuclear parton distribution functions. At the LHC, the pair-production cross sections amount to σtt¯ = 3.4 μb in Pb–Pb at √sNN = 5.5 TeV, and σtt¯ = 60 nb in p–Pb at √sNN = 8.8 TeV. At the FCC energies of √sNN = 39 and 63 TeV, the same cross sections are factors of 90 and 55 times larger respectively. In the leptonic final-state tt¯ → W +b W − ¯ b → b ¯ b νν with = e±,μ±, after typical acceptance and efficiency cuts, one expects about 90 and 300 top-quarks per nominal LHC-year and 4.7 · 104 and 105 per FCC-year in Pb–Pb and p–Pb collisions respectively. The total tt¯ cross sections, dominated by gluon fusion processes, are enhanced by 3–8% in nuclear compared to p–p collisions due to an overall net gluon antishadowing, although different regions of their differential distributions are depleted due to shadowing or EMC-effect corrections. The rapidity distributions of the decay leptons in tt¯ processes can be used to reduce the uncertainty on the Pb gluon density at high virtualities by up to 30% at the LHC (full heavy-ion programme), and by 70% per FCC-year. The cross sections for single-top production in electroweak processes are also computed, yielding about a factor of 30 smaller number of measurable top-quarks after cuts, per system and per year.