This paper investigates an extension of the May-Nowak ODE model for virus dynamics with gradient-dependent flux limitation of cross diffusion. In particular, we consider the associated no-flux initial–boundary value problem (0.1)ut=Δu−∇⋅(uf(|∇v|2)∇v)+κ−u−uw,vt=Δv−v+uw,wt=Δw−w+vin a smoothly bounded domain Ω⊂Rn(n≤3), where the parameter κ≥0. The prototypical chemotactic sensitivity function f∈C2([0,∞)) is given by f(ξ)=(1+ξ)−α,ξ≥0 with some α∈R. It is proved that whenever α∈R,ifn=1,α>n−22(n−1),ifn={2,3},global classical solutions to (0.1) exist and are uniformly bounded. Such result consists with that in [Winkler (2022), Proposition 1.2] when n≤3, which shows that the effect of gradient-dependent flux limitation in weakening the cross-diffusion term remains unchanged even in the context of nonlinear signal production mechanism.