The energy spectrum of an electron in a two-dimensional concentric double quantum ring is exactly calculated in the presence of an axial magnetic field using linear combinations of confluent hypergeometric functions. The influence of the geometric confinement on the energy spectrum is analyzed as a function of the inner and outer ring radii. The applied magnetic field modifies the electron energy spectrum and we discuss the appearance of the well-known Aharonov–Bohm oscillations. Interesting behavior of the probability amplitude of the electron state was found reflecting the competition between geometric and magnetic confinements. Electronic tunneling between the coupled rings may be reinforced or suppressed by conveniently modulating the barrier size separating the two rings, leading to drastic change in the electronic charge distribution through the nanostructure.