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.