The electrical conductance G of magnetostrictive nanocontacts made from Galfenol (Fe73Ga27) can be reproducibly switched between "on" and "off" states in a low magnetic field of ∼20-30 mT at 10 K. The switching behavior is in agreement with the magnetic field dependence of the magnetostriction inferred from the magnetization behavior, causing a positive magnetostrictive strain along the magnetic field. The repeated magnetic-field cycling leads to a stable contact geometry and to a robust contact configuration with a very low hysteresis of ∼1 mT between opening and closing the contact due to a training effect. Non-integral multiples of the conductance quantum G0 observed for G > G0 are attributed to electron backscattering at defect sites in the electrodes near the contact interface. When the contact is closed either mechanically or by magnetic field, the conductance shows an exponential behavior below G0 due to electron tunneling. This allows to estimate the magnetostriction λ = 4 × 10-5 at 10 K. The results demonstrate that such magnetostrictive devices are suitable for the remote control of the conductance by low magnetic fields in future nanotechnology applications. © 2016 Author(s).