  We study relations between M-strings (one-dimensional intersections of
M2-branes and M5-branes) in six dimensions and m-strings (magnetically charged
monopole strings) in five dimensions. For specific configurations, we propose
that the counting functions of BPS bound-states of M-strings capture the
elliptic genus of the moduli space of m-strings. We check this proposal for the
known cases, the Taub-NUT and Atiyah-Hitchin spaces for which we find complete
agreement. Furthermore, we analyze the modular properties of the M-string free
energies, which do not transform covariantly under SL(2,Z). However, for a
given number of M-strings, we find that there exists a unique combination of
unrefined genus-zero free energies that transforms as a Jacobi form under a
congruence subgroup of SL(2,Z). These combinations correspond to summing over
different numbers of M5-branes and make sense only if the distances between
them are all equal. We explain that this is a necessary condition for the
m-string moduli space to be factorizable into relative and center-of-mass
parts.
