  Let $\mathbb{F}$ be a finite field consisting of $q$ elements and let $n \geq
1$ be an integer. In this paper, we study the size of local Kakeya sets with
respect to subsets of $\mathbb{F}^{n}$ and obtain upper and lower bounds for
the minimum size of a (local) Kakeya set with respect to an arbitrary set
${\mathcal T} \subseteq \mathbb{F}^{n}.$
