  We consider the asymptotic expansion of the Mathieu-Bessel series
\[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu},
\qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other
parameters held fixed, where $J_\nu(x)$ is the Bessel function of the first
kind of order $\nu$. A special case arises when $\gamma+\nu$ is a positive even
integer, where the expansion comprises finite algebraic terms together with an
exponentially small expansion. Numerical examples are presented to illustrate
the accuracy of the various expansions. The expansion of the alternating
variant of $S_\nu(a,b)$ is considered. The series when the $J_\nu(x)$ function
is replaced by the Bessel function $Y_\nu(x)$ is briefly mentioned.
