Electron transition rates due to interaction with circularly polarized light incident along the axis of a free-standing solid cylindrical nanowire are evaluated in the dipole approximation. The electric confinement potential of the nanowire is modeled as a superposition of two parts, in general, of different strengths; viz; parabolic and inverse parabolic in the radial distance. Additional confinement of the charge carriers is through the vector potential of the axial applied magnetic field. In systems with cylindrical symmetry, the electronic states are in part characterized by azimuthal quantum numbers: m=0, ±1, ±2,..., which in the absence of the axial applied magnetic field are doubly degenerate. In the dipole approximation and for circularly polarized light the selection rules are such that optical transitions are allowed between electronic states whose azimuthal quantum numbers differ by unity. Transition rates are characterized by peaks whenever the energy of the incident electromagnetic radiation matches transition energies for states between which transitions occur. The parabolic potential blue shifts peak of transition rates while the inverse parabolic potential redshifts the peaks. Results also indicate that transition rates are higher in nanowires of smaller radii. The homogeneous magnetic field lifts the double-degeneracies of electrons with opposite angular momenta, which leads to the emergence of two branches of the transition rates.