Lead is an important alloy material nuclide. And lead eutectic is also an important coolant, which is applied in the construction of Lead-cooled Fast Reactor such as The European Lead-cooled System (ELSY) and the China Lead-based Research reactor (CLEAR-I), as well as in research related to Generation-IV reactor. The study and calculation of lead nuclear data have important theoretical value and application prospects. ²⁰⁸Pb is the most stable and abundant isotope in lead nuclei, and high quality description of ²⁰⁸Pb nuclear scattering data is the key to achieving theoretical calculations of nuclear reaction data for lead nuclei. Based on the dispersive optical model, this work describes nucleon scattering on ²⁰⁸Pb by using the dispersive optical potential. The dispersive optical model potential is defined by energy-dependent real potentials, imaginary potentials, the corresponding dispersive contributions to the real potential which are calculated analytically from the corresponding imaginary potentials by using a dispersion relation, and isospin dependence is reasonably considered by introducing isovector component (i.e. Lane term) in the potential depth constants of the real Hartree-Fock potential $ V_{\rm{HF}}$ and the surface imaginary potential $ W_{\rm{s}}$. Unlike K-D potential, which requires two different sets of parameters to describe neutron and proton induced scattering data, this optical potential uses the same set of parameters to simultaneously describe nucleon-nucleus scattering data. The derived potential in this work shows a very good description of nucleon-nucleus scattering data on ²⁰⁸Pb up to 200 MeV. Calculated neutron total cross sections, neutron and proton elastic scattering angular distributions, as well as neutron and proton elastic analyzing powers are shown to be in good agreement with experimental data. Additionally, the difference in potential between the neutron and protons induced is described by the isovector term, a reasonable good prediction of quasielastic (p, n) scattering data is achieved.