In this paper $i$-permutable $n$-groupoids are defined and considered. An $n$-groupoid $(G,f)$ is called $i$-permutable iff $f(x_1,\dots,x_{i-1},f(x_1,\dots,x_n),x),x_i,\dots,x_{n-1})=x_n$ for all $x_i,\dots,x_n\in G$ and fixed $i\in\{1,\dots,n\}$. $i$-permutable $n$-groupoids represent a generalization of several classes of binary and $n$-ary groupoids: semisymmetric groupoids, groupoids satisfying Sade's left "key's" law and cyclic $n$-groupoids. Examples, of $i$-permutable $n$-groupoids are given and some properties of such $n$-groupoids described, $i$-permutable $n$-groupoids satisfyng some commutativity and associativity conditions are studied. Several conditions for an $i$-permutable $n$-groupoid to be an $n$-group are determined.