Resistance and current-voltage characteristics of individual superconducting NbSe2 nanowires are investigated. In the current-voltage curves, a stairlike structure is observed, indicating the possible formation of phase-slip centers. A close examination of the current-voltage characteristic in a selected high quality NbSe2 nanowire with a diameter of 75 nm reveals that the characteristic voltages in the stairlike structure follow the BCS-like temperature dependence of superconducting gaps vanishing at TC. While the phase-slip center mechanism remains to be a plausible explanation of the observed features, an alternative model involving multigap Josephson tunneling is proposed to account for the BCS-like temperature dependence. From the BCS fits, two distinct superconducting gaps are extracted. Moreover, the critical current of the 75 nm nanowire at low temperatures as well as near TC can also be described by the Ambegaokar-Baratoff relation for multigap Josephson junctions. Our data suggest the possible observation of multiband superconductivity in NbSe2 and are in good agreement with the predictions of recent band structure and Fermi surface calculations on NbSe2.