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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 uniquerepresentation in the form $\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$. Let$D$ be a subset of $\mathbb{F}_p$. We consider the set $W_D$ of elements of$\mathbb{F}_q$ such that $c_j\in D$ for all $j=1,\ldots,r$. We give an estimatefor the number of squares in the set $W_D$ which implies a weaker sufficientcondition for the existence of squares in the set $W_D$ than in the recentpaper of C.Dartyge, C.Mauduit, A.S\'ark\"ozy.

On squares in subsets of finite fields with restrictions on coefficients of basis decomposition