Let \(n\geq 2\) and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Suppose that \(\Omega\) is a homogeneous function of degree zero on \(\mathbb{R}^n\) that satisfies \(\Omega\in L(S^{n-1})\) and \(\int_{S^{n-1}}\Omega\,d\sigma= 0\). Define the Marcinkiewicz integral operator \(\mu_\Omega\) by \[ \mu_\Omega(f)(x)= \Biggl(\int^\infty_0\,\Biggl|\, \int_{| x-y| 1/2\), \[ \sup_{|\zeta|=1}\, \int_{S^{n-1}} |\Omega(\theta)|(\log_2|\langle \theta,\zeta\rangle|^{- 1})^\alpha\, d\theta< \infty, \] then the operator \(\mu_\Omega\) is bounded on \(L^p(\mathbb{R}^n)\) for \(4\alpha/(4\alpha- 1)< p< 4\alpha\). The above condition on \(\Omega\) in the theorem was introduced by \textit{L. Grafakos} and \textit{A. Stefanov} [Indiana Univ. Math. J. 47, No. 2, 455--469 (1998; Zbl 0913.42014)]. The case \(\alpha= 1/2\) and its relation with the \(L(\log^+L)\) spaces can be found in [\textit{T. Walsh}, Stud. Math. 44, 203--217 (1972; Zbl 0212.13603)].