The production potential of the excited neutrinos at the FCC-based electron-hadron colliders, namely, the ERL60⊗FCC with s=3.46 TeV, the ILC⊗FCC with s=10 TeV, and the PWFA-LC⊗FCC with s=31.6 TeV, has been analyzed. The branching ratios of the excited neutrinos have been calculated for the different decay channels and shown that the dominant channel is ν⋆→eW+. We have calculated the production cross sections with the process of ep→ν⋆q→eW+q and the decay widths of the excited neutrinos with the process of ν⋆→eW+. The signals and corresponding backgrounds are studied in detail to obtain accessible mass limits. It is shown that the discovery limits obtained on the mass of the excited neutrino are 2452 GeV for Lint=100 fb-1, 5635 GeV for Lint=10 fb-1 (6460 GeV for Lint=100 fb-1), and 10200 GeV for Lint=1 fb-1 (13960 GeV for Lint=10 fb-1), for the center-of-mass energies of 3.46, 10, and 31.6 TeV, respectively.