  Let $G$ be a finite $p$-group. In this paper we obtain bounds for the
exponent of the non-abelian tensor square $G \otimes G$ and of $\nu(G)$, which
is a certain extension of $G \otimes G$ by $G \times G$. In particular, we
bound $\exp(\nu(G))$ in terms of $\exp(\nu(G/N))$ and $\exp(N)$ when $G$ admits
some specific normal subgroup $N$. We also establish bounds for $\exp(G \otimes
G)$ in terms of $\exp(G)$ and either the nilpotency class or the coclass of the
group $G$, improving some existing bounds.
