We present a catalog of 908 objects observed with the Large Sky Area MultiObject Fiber Spectroscopic Telescope (LAMOST) in the vicinity fields of M31 and M33, targeted as globular clusters (GCs) and candidates. The targets include known GCs and candidates selected from the literature, as well as new candidates selected from the Sloan Digital Sky Survey (SDSS). Analysis shows that 356 of them are likely GCs of various degree of confidence, while the remaining ones turn out to be background galaxies and quasars, stars and H II regions in M31 or foreground Galactic stars. The 356 likely GCs include 298 bona fide GCs and 26 candidates known in the literature. Three candidates selected from the Revised Bologna Catalog of M31 GCs and candidates (RBC) and one possible cluster from Johnson et al. are confirmed to be bona fide clusters. We search for new GCs in the halo of the M31 amongst the new candidates selected from the SDSS photometry. Based on radial velocities yielded by LAMSOT spectra and visual examination of the SDSS images, we find 28 objects, 5 bona fide and 23 likely GCs. Amongst the five bona fide GCs, three have been recently discovered independently by others, the remaining 25 are our new identifications, including two bona fide ones. The new identified objects fall at projected distances ranging from 13 to 265 kpc from M31. Of the two newly discovered bona fide GCs, one is located near M33, probably a GC belonging to M33. The other bona fide GC falls on the Giant Stream with a projected distance of 78 kpc from M31. Of the 23 newly identified likely GCs, one has a projected distance of about 265 kpc from M31 and could be an intergalactic cluster.