We report the detection of a transiting, dense Neptune planet candidate orbiting the bright (V=8.6) K0.5V star HD 95338. Detection of the 55-d periodic signal comes from the analysis of precision radial velocities from the Planet Finder Spectrograph on the Magellan II Telescope. Follow-up observations with HARPS also confirm the presence of the periodic signal in the combined data. HD 95338 was also observed by the Transiting Exoplanet Survey Satellite (TESS) where we identify a clear single transit in the photometry. A Markov chain Monte Carlo period search on the velocities allows strong constraints on the expected transit time, matching well the epoch calculated from TESS data, confirming both signals describe the same companion. A joint fit model yields an absolute mass of 42.44^+2.22^_-2.08_M_{Earth}_ and a radius of 3.89^+0.19^_-0.20_R_{Earth}_, which translates into a density of 3.98^+0.62^_-0.64_g/cm^3^ for the planet. Given the planet mass and radius, structure models suggest it is composed of a mixture of ammonia, water, and methane. HD 95338 b is one of the most dense Neptune planets yet detected, indicating a heavy element enrichment of ~90 per cent (~38M_{Earth}_). This system presents a unique opportunity for future follow-up observations that can further constrain structure models of cool gas giant planets.