This theoretical study deals with the Navier–Stokes equations posed in a 3D thin domain with thickness $$0<\varepsilon \ll 1$$, assuming power-law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced in (Djoko et al. Comput. Math. Appl. 128 (2022) 198–213), represents a generalization of the Navier slip boundary condition. The goal is to study the influence of the power-law slip boundary conditions with an anisotropic tensor of order $$\varepsilon ^{\gamma \over s}$$, with $$\gamma \in \mathbb {R}$$ and flow index $$1<s<2$$, on the behavior of the fluid with thickness $$\varepsilon$$ by using asymptotic analysis when $$\varepsilon \rightarrow 0$$, depending on the values of $$\gamma$$. As a result, we deduce the existence of a critical value of $$\gamma$$ given by $$\gamma _s^*=3-2s$$ and so, three different limit boundary conditions are derived. The critical case $$\gamma =\gamma _s^*$$ corresponds to a limit condition of type power-law slip. The supercritical case $$\gamma >\gamma _s^*$$ corresponds to a limit boundary condition of type perfect slip. The subcritical case $$\gamma <\gamma _s^*$$ corresponds to a limit boundary condition of type no-slip.