Summary: Let \(1<r<\infty, B_j\) and \(\phi_j(j=1, \ldots, m)\) are suitable functions. We say a kernel \(K\) satisfies a variant of the classical \(L^r\)-Hörmander’s condition, if there exist constants \(c_r>0\) and \(C_r>0\), such that for any \(y\in \mathbb R^n\) and \(R>c_r|y|\), \[ \sum\limits^\infty_{k=1}(2^kR)^{n/r'}\left(\int_{2^kR<|x|\leq 2^{k+1}R}\left|K(x-y)-\sum\limits^m_{j=1}B_j(x)\phi_j(y)\right|^rdx\right)^{1/r}\leq C_r. \] In this paper, the authors establish the weighted norm inequalities for singular integral operators with kernel satisfying the above variant of the classical \(L^r\)-Hörmander’s condition.