Using a deep (similar to 20,200 s) ROSAT PSPC image we have examined the central region of the Coma cluster. Two extended regions of enhanced X-ray emission are found, centered at the positions of the brightest elliptical galaxies in the cluster: NGC 4874 and NGC 4889. Spectral analysis of the sources reveals no evidence of any difference between the spectra of these sources and that of the surrounding cluster emission. We assume that the enhancement in the X-ray surface brightness results from gas density enhancements and also that the underlying mass concentrations lie either at the cluster center or 1 core radius out of the center (420 kpc). With these assumptions, we derive total masses of 1.2 x 10(13)-1.6 x 10(13) and 0.9 x 10(13)-1.8 x 10(13) M. within 2' (80 kpc) of NGC 4874 and NGC 4889, respectively, assuming a Hubble constant H-0 = 50 km s Mpc(-1). Corresponding mass-to-light ratios for the galaxies are 30-40 and 25-50 in solar units, increasing at larger radii and approaching the values derived for the entire cluster at distances of more than similar to 150 kpc from the galaxies.