To study the supercooling of many liquid metals with high melting temperatures, the investigation was performed for the cooling curves of electrostatic levitation (ESL) experiments, which had been originally obtained for the measurements of many physical properties. The largest supercooling over the literature values was found for liquid Ru (428 K), Ta (721 K), W (601 K), and Ir (438 K), where temperatures in the parentheses mean the supercooling of respective liquid metals. This indicates the validity of ESL for the supercooling experiments of liquid metals and alloys because of being rather free from the heterogeneous nucleation. This ESL was applied to the study of supercooling of homogeneous liquid phase in the composition range from 29 at% Nb to 71 at% Nb of eutectic Ni-Nb system, whose eutectic point is present at 40.5 at% Nb and 1448 K. The experimental result shows a poor supercooling tendency of homogeneous liquid phase around the eutectic composition in spite of large supercooling far apart from this eutectic composition. This characteristic feature was discussed based on the classical nucleation theory coupled with the knowledge recently found, the existence of concentration fluctuations in the homogeneous liquid phase near the eutectic point.