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Strong nonlocality with genuine entanglement was first shown by Wang \emph{et al.} using sets of GHZ-like states in tripartite quantum systems [Phys. Rev. A \textbf{104}, 012424 (2021)]. However, it is an open problem whether there exists strong nonlocality with genuine entanglement in four or more partite systems. In this paper, we unify two different concepts of strong nonlocality introduced by Halder \emph{et al.} [Phys. Rev. Lett. \textbf{122}, 040403 (2019)] and by Zhang \emph{et al.} [Phys. Rev. A \textbf{99}, 062108 (2019)]. That is, we use a concept of $k$-strong nonlocality instead of these two different types of strong nonlocality. A set of orthogonal quantum states is $k$-strong nonlocal if it is locally irreducible in every $k$-partition. In fact, the strong nonlocality that is usually said is 2-strong nonlocality. The smaller the $k$ is, the stronger the nonlocality will be. A set of states is $k_{+}$-strong nonlocal if the strong nonlocality of this set is stronger than $k$-strong nonlocality but weaker than $(k-1)$-strong nonlocality. Based on these concepts, firstly, we show 2-strong nonlocality with genuine entanglement by some sets of GHZ-like states with weight $d$ in tripartite systems. These sets are not necessarily complete bases. Secondly, we present 2-strong nonlocality with genuine entanglement for systems with four or more parties. These results solve the open problem raised by Wang \emph{et al.} Finally, we construct a set of GHZ-like states with $n_+$-strong nonlocality in $n$-partite quantum systems.