We calculate the binding energy and the density of states of a hydrogenic shallow donor impurity in quantum-well wires with rectangular cross section using a variational procedure within the effective-mass approximation under the influence of a constant electric field applied perpendicular to the axes of the wire. Calculations were performed as functions of the position of the impurity in a quantum-well wire of infinite depth and for various sizes of the wire cross section. The binding energy and the density of impurity states are analysed and compared with their quantum-well wire counterparts. Our results indicate that a proper consideration of the density of impurity states when electric fields are applied in the structure may be of relevance in the interpretation of future experimental data on optical phenomena related to shallow impurities in quantum-well wires.