In this article, we consider the Marcinkiewicz integrals with variable kernels defined by μΩ(f)(x)=(0||xy|tΩ(x,xy)|xy|n1f(y)dy|2dtt3)1/2, where Ω(x,z)L(n)×Lq(Sn1) for q > 1. We prove that the operator μΩ is bounded from Hardy space, Hp(n), to Lp(n) space; and is bounded from weak Hardy space, Hp,(n), to weak Lp(n) space for max{2n2n+1,nn+α}<p<1, if Ω satisfies the L1,α-Dini condition with any 0<α1.