ABSTRACT Observations of Einstein LMC X-Ray point sources have been made with ROSAT's High Resolution Imager to obtain accurate positions from which to search for optical counterparts. This paper is the first in a series reporting results of the ROSAT observations and subsequent optical observations. It includes the X-ray positions and fluxes, information about variability, optical finding charts for each source, a list of identified counterparts, and information about candidates which have been spectroscopically observed in each of the fields. Sixteen point sources were measured at a a>3-sigma level, which 15 other sources were either extended or less significant detections. About 50% of the sources are serendipitous detections (not found in previous surveys). More than half of the X-ray sources are variable. Sixteen of the sources have been optically identified or confirmed: 6 with foreground cool stars, 4 with Seyfert galaxies, 2 with SNR in the LMC, and 4 with peculiar hot LMC stars. Presumably the latter are all binaries, although only one (CAL 83) has been previously studied in detail.