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 ${\mathcal T} \subseteq \mathbb{F}^{n}.$