  We consider a superconducting film exchange-coupled to a close-by chiral
magnetic layer and study how magnetic skyrmions can induce the formation of
Majorana bound states (MBS) in the superconductor. Inspired by a proposal by
Yang et al. [Phys. Rev. B 93, 224505 (2016)], which suggested MBS in skyrmions
of even winding number, we explore whether such skyrmions could result from a
merger of ordinary skyrmions. We conclude that the formation of higher-winding
skyrmions is not realistic in chiral magnets. Subsequently, we present a
possibility to obtain MBS from realistic skyrmions of winding number one, if a
skyrmion-vortex pair is formed instead of a bare skyrmion. Specifically, we
show that MBS are supported in a pair of a circular skyrmion and a vortex which
both have a winding number of one. We back up our analytical prediction with
results from numerical diagonalization and obtain the spatial profile of the
MBS. In light of recent experimental progress on the manipulation of skyrmions,
such systems are promising candidates to achieve direct spatial control of MBS.
