Using the effective-mass approximation within the variational approach, we have calculated the binding and the allowed transition energies between the $1s$-, $2s$-, ${2p}_{\ifmmode\pm\else\textpm\fi{}}$-, ${2p}_{z}$-, $3s$-, ${3p}_{\ifmmode\pm\else\textpm\fi{}}$-, and ${3p}_{z}$-like states of a hydrogenic on-center donor impurity in a cylindrical GaAs quantum-well wire, under the action of a magnetic field applied in the axial direction. Our results are obtained for several wire radii and as a function of the applied magnetic field. We have found that some excited states are not bounded for some values of the radius of the wire and of the applied magnetic field, and that the geometric confinement and the applied magnetic field split the degeneracy of some excited states.