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Here we give some approximation theorems concerning pointwise convergence and rate of pointwise convergence of nonlinear singular integral operators with non-isotropic kernels of the form: $T_{w,λ}(f)(s) =\int_{R^n} K_w (|s − t|_\lambda, f(t)) dt$, with kernels satisfying Lipschitz and some singularity assumptions. Here $\Lambda$ is a non-empty set of indices, $w_0$ is an accumulation point of $\Lambda$ and $|s − t|_\lambda$ denotes the non-isotropic distance between the points $s, t \in R^n$.