The authors prove the existence of positive radial solutions (decaying to zero as \(|x|\to\infty\)) of the quasilinear system \(-\text{div} (A_i(|\nabla u_i|)\nabla u_i)=\sum_j a_{ij}(|x|)f_{ij}(u_j)\), \(x\in\mathbb R^N\), \(N>1\), \(i=1,2,\dots,n\). The functions \(A_i,a_{ij},f_{ij}\) are continuous, \(A_i>0\), \(a_{ij}\geq 0\), \(f_{ij}(0)=0\), \(f_{ij}(s)>0\) for \(s>0\), \(f_{ij}(s)\to\infty\) as \(s\to\infty\). Moreover, the functions \(A_i,f_{ij}\) are supposed to be asymptotically homogeneous at zero and at infinity (with suitable powers satisfying several inequalities) and an integrability condition involving \(A_i,a_{ij}\) has to be satisfied. The proof is based on the use of the Leray-Schauder degree and on a modification of a priori estimates of \textit{B. Gidas} and \textit{J. Spruck} [Commun. Partial Differ. Equations 6, 883-901 (1981; Zbl 0462.35041)]. Two examples with nonlinearities \(A_i,f_{ij}\) of the form \(s^p\log(1+s)\), \(s^p\) and \(\log(1+s^p)/\log(1+s^{-q})\) are given.