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We consider the linear vector space formed by the elements of the finitefields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let$\{a_1,\ldots,a_r\}$ be a basis of this space. Then the elements $x$ of$\mathbb{F}_q$ have a unique representation in the form $\sum_{j=1}^r c_ja_j$with $c_j\in\mathbb{F}_p$. Let $D_1,\ldots,D_r$ be subsets of $\mathbb{F}_p$.We consider the set $W=W(D_1,\ldots,D_r)$ of elements of $\mathbb{F}_q$ suchthat $c_j \in D_j$ for all $j=1,\ldots,r$. We give an estimate for the numberof squares in the set $W$ which implies an asymptotic formula for this quantityin the case when the sets $D_1,\ldots,D_r$ are "large on average" and asufficient condition for the existence of squares in the set $W$.

On squares in special sets of finite fields