The binding energy of an off-axis shallow-donor impurity in a GaAs coaxial cylindrical quantum well-wire under the action of an intense, high-frequency laser field is calculated using a variational procedure within the effective-mass approximation. We take into account the laser dressing effects on both the impurity Coulomb potential and the confinement potential. Numerical calculations of the ground state subband energy levels based on a finite element method are performed for different barrier thicknesses and laser field parameters. Our model indicates a possible tuning of the well potential shape along the laser polarization direction. Strong distortions of the electron probability density under intense laser field conditions are also predicted. The study proves that the presence of the laser field partially breaks down the degeneracy of the states for donors symmetrically positioned within the structure. The results obtained show that the impurity energy levels in coaxial quantum wires can be significantly modified and controlled by intense laser fields.