In this paper $\sigma$-permutable $n$-groupoids are defined and considered. An $n$-groupold $(G,f)$ is called $\sigma$-permutable, where $\sigma$ is a permutation of the set $\{1,\dots,n+1\}$, iff $f(x_{\sigma_1},\dots,x_{\sigma_n})=x_{\sigma(n+1)}\Leftrightarrow f(x_1,\dots,x_n)=x_{n+1}$ for all $x_1,\dots,x_{n+1}\in G$. $\sigma$-permutable $n$-groupoids are a generalization of several classes of $n$-groupoids. Examples of $\sigma$-permutable $n$-groupoids are given and some of their properties described. Several conditions under which $\sigma$-permutable $n$-groupoids are $n$-groups are determined.