We report on the measurements of the superconducting order parameter in the nonmagnetic borocarbides LuNi2B2C and YNi2B2C. Andreev conductance spectra are obtained from nanoscale metallic junctions on single crystal surfaces prepared along three major crystallographic orientations: [001], [110], and [100]. The gap values extracted by the single-gap Blonder-Tinkham-Klapwijk model follow the theoretical predictions as a function of temperature and magnetic field and exhibit a small anisotropy with no indication of proposed gap nodes along the [100] and [010] directions. These observations are robust and reproducible among all the measurements on two different sets of LuNi2B2C crystals and one set of YNi2B2C crystals. We suggest that the possible gap nodes in the [100] direction may be masked by two effects: different gap anisotropy across multiple Fermi surfaces, as reported in the recent photoemission spectroscopy, and the large tunneling cone. Our results provide a consistent picture of the superconducting gap structure in these materials, addressing the controversy particularly in the reported results of point-contact Andreev reflection spectroscopy.