The magnetic and magnetotransport properties of metallic 1$T$-VTe$_{2}$ single crystals were investigated at temperatures from 1.3 to 300 K and in magnetic fields up to 35 T. Upon applying a high magnetic field, it is found that the electrical resistivity displays a crossover from the logarithmic divergence of the single-impurity Kondo effect to the Fermi liquid behavior at low temperatures. The Brillouin scale of the negative magnetoresistivity above the Kondo temperature $T_{rm{K}}$ = 12 K indicates that the Kondo features originate from intercalated V ions, with $S$ = 1/2. Both magnetic susceptibility and Hall effect show an anomaly around $T_{rm{K}}$. By using the modified Hamann expression we successfully describe the temperature-dependent resistivity under various magnetic fields, which shows the characteristic peak below $T_{rm{K}}$ due to the splitting of the Kondo resonance.
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