The LUCIFER project aims at deploying the first array of enriched scintillating bolometers for the investigation of neutrinoless double-beta decay of Se-82. The matrix which embeds the source is an array of ZnSe crystals, where enriched Se-82 is used as decay isotope. The radiopurity of the initial components employed for manufacturing crystals, that can be operated as bolometers, is crucial for achieving a null background level in the region of interest for double-beta decay investigations. In this work, we evaluated the radioactive content in 2.5 kg of 96.3 % enriched Se-82 metal, measured with a high-purity germanium detector at the Gran Sasso deep underground laboratory. The limits on internal contaminations of primordial decay chain elements of Th-232, U-238 and U-235 are respectively: <61,<110 and <74 mu Bq/kg at 90 % C.L. The extremely low-background conditions in which the measurement was carried out and the high radiopurity of the Se-82 allowed us to establish the most stringent lower limits on the half-lives of the double-beta decay of Se-82 to 0(+)(1), 2(2)(+) and 2(1)(+) excited states of Kr-82 of 3.4.10(22), 1.3.10(22) and 1.0.10(22) y, respectively, with a 90 % C.L.