  Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $\exp(G)$
and let $m=\lfloor\log_pk\rfloor$. We show that $\exp(M^{(c)}(G))$ divides
$\exp(G)p^{m(k-1)}$, for all $c\geq1$, where $M^{(c)}(G)$ denotes the
c-nilpotent multiplier of $G$. This implies that $\exp(M(G))$ divides $\exp(G)$
for all finite $p$-groups of class at most $p-1$. Moreover, we show that our
result is an improvement of some previous bounds for the exponent of
$M^{(c)}(G)$ given by M. R. Jones, G. Ellis and P. Moravec in some cases.
