In recent years, the methods of severe plastic deformation and rapid melt quenching have proven to be an effective tool for the formation of the unique properties of materials. The effect of high-pressure torsion (HPT) on the structure of the amorphous alloys of the quasi-binary TiNi–TiCu system with a copper content of more than 30 at.% produced by melt spinning technique has been analyzed using the methods of scanning electron microscopy, X-ray diffraction analysis, and differential scanning calorimetry (DSC). The structure examinations have shown that the HPT of the alloys with a Cu content ranging from 30 to 40 at.% leads to nanocrystallization from the amorphous state. An increase in the degree of deformation leads to a substantial change in the character of the crystallization reflected by the DSC curves of the alloys under study. The alloys containing less than 34 at.% Cu exhibit crystallization peak splitting, whereas the alloys containing more than 34 at.% Cu exhibit a third peak at lower temperatures. The latter effect suggests the formation of regions of possible low-temperature crystallization. It has been established that the HPT causes a significant decrease in the thermal effect of crystallization upon heating of the alloys with a high copper content relative to that of the initial amorphous melt quenched state.