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), 10.1103/PhysRevB.93.224505], 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.