  We consider the asymptotic expansion of the Mathieu-Bessel series
\[S_{\nu,\gamma}^{\mu}(a,b)=\sum_{n=1}^\infty \frac{n^\gamma
K_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu>0, \nu\geq 0, b>0, \gamma\in {\bf
R})\] as $|a|\to\infty$ in $|\arg\,a|<\pi/2$ with the other parameters held
fixed, where $K_\nu(x)$ is the modified Bessel function of the second kind of
order $\nu$. We employ a Mellin transform approach to determine an integral
representation for $S_{\nu,\gamma}^{\mu}(a,b)$ involving the Riemann zeta
function. Asymptotic evaluation of this integral involves appropriate residue
calculations. Numerical examples are presented to illustrate the accuracy of
each type of expansion obtained. The expansion of the alternating variant of
$S_{\nu,\gamma}^\mu(a,b)$ is also considered.
