Summary: In this article, we study the existence and multiplicity of positive solutions for the Neumann boundary value problems involving the \(p(x)\)-Kirchhoff of the form \[ \begin{cases} -M\left(\int_\Omega \frac{1}{p(x)}\left(|\nabla u|^{p(x)}+\lambda|u|^{p(x)}\right)dx\right)\left(\mathrm{div}\left(|\nabla u|^{p(x)-2}\nabla u\right)-\lambda|u|^{p(x)-2}u\right)=f(x,u) & \;\text{in} \;\Omega, \\ \frac{\partial u}{\partial v}=0 & \;\text{on} \;\partial \Omega.\end{cases} \] Using the sub-supersolution method and the variational method, under appropriate assumptions on \(f\) and \(M\), we prove that there exists \(\lambda^\ast > 0\) such that the problem has at least two positive solutions if \(\lambda > \lambda^\ast\), at least one positive solution if \(\lambda = \lambda^\ast\) and no positive solution if \(\lambda < \lambda^\ast\). To prove these results we establish a special strong comparison principle for the Neumann problem.