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 observed spectroscopically in each of the fields. Sixteen point sources were measured at a >3sigma level, while 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: six with foreground cool stars, four with Seyfert galaxies, two with SNR in the LMC, and four with peculiar hot LMC stars. Presumably the latter are all binaries, although only one (CAL 83) has been previously studied in detail.