The present status of searches for the neutrino magnetic moment (NMM) down to 3×10−11 μB is briefly discussed. The low background Ge-NaI spectrometer GEMMA constructed in ITEP for NMM measurement is described and the results of the test background measurements at a small depth of 5 m.w.e in ITEP are reported. The analysis of these results shows that the limit on NMM at the level (2 ÷ 3)·10−11 μB can be achieved during 2 year duration of measurements with spectrometer GEMMA at Kalininskaya Nuclear Power Plant (KNPP). The prospects of breakthrough to the limit on NMM about 10−12 μB are outlined.