A powder sample and single-crystal plates of the topological insulator Bi₂Te₃ have been investigated using the ¹²⁵Te NMR method at room temperature and at low temperatures in the range from 12.5 to 16.5 K. The NMR spectra of the single-crystal plates have been studied in the orientation where the crystallographic axis c is directed parallel or perpendicular to the magnetic field. The spectra have been obtained by means of recording spin-echo signals and plotting their envelopes. It has been shown that the NMR spectra for the bismuth telluride powder and plates with the orientation c ⊥ B consist of two lines, which are presumably attributed to tellurium nuclei in two crystallographic positions in the bulk of the sample. The position and shape of the lines are determined by the chemical shift and the Knight shift. For the orientation of the plates c || B, the spectrum contains an additional component in the high-frequency region, which cannot appear due to the angular dependence of the line shifts caused by tellurium nuclei in the bulk of the topological insulator. At a low temperature, the additional line dominates in the spectrum.