I report the discovery of a transient broad-H$alpha$ point source in the outskirts of the giant elliptical galaxy NGC 1404, discovered in archival observations taken with the MUSE integral field spectrograph. The H$alpha$ line width of 1950 km s$^{-1}$ FWHM, and luminosity of (4.1$pm$0.1)$times$10$^{36}$ erg s$^{-1}$, are consistent with a nova outburst, and the source is not visible in MUSE data obtained nine months later. A transient soft X-ray source was detected at the same position (within $<$1 arcsec), 14 years before the H$alpha$ transient. If the X-ray and H$alpha$ emission are from the same object, the source may be a short-timescale recurrent nova with a massive white dwarf accretor, and hence a possible Type-Ia supernova progenitor. Selecting broad-H$alpha$ point sources in MUSE archival observations for a set of nearby early-type galaxies, I discovered twelve more nova candidates with similar properties to the NGC 1404 source, including five in NGC 1380 and four in NGC 4365. Multi-epoch data are available for four of these twelve sources; all four are confirmed to be transient on $sim$1 year timescales, supporting their identification as novae.
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