Inverse Thomson scattering is a well-known radiation process that produces high-energy photons both in nature and in the laboratory. Nonlinear inverse Thomson scattering occurring inside an intense light field is a process which generates higher harmonic photons. In this paper, we theoretically show that the higher harmonic gamma-ray produced by nonlinear inverse Thomson scattering of circularly polarized light is a gamma-ray vortex, which means that it possesses a helical wave front and carries orbital angular momentum. Our work explains a recent experimental result regarding nonlinear inverse Thomson scattering that clearly shows an annular intensity distribution as a remarkable feature of a vortex beam. Our work implies that gamma-ray vortices should be produced in various situations in astrophysics in which high-energy electrons and intense circularly polarized light fields coexist. Nonlinear inverse Thomson scattering is a promising radiation process for realizing a gamma-ray vortex source based on currently available laser and accelerator technologies, which would be an indispensable tool for exploring gamma-ray vortex science.