Abstract We present the first X-ray reflection model for testing the assumption that the metric of astrophysical black holes is described by the Kerr solution. We employ the formalism of the transfer function proposed by Cunningham. The calculations of the reflection spectrum of a thin accretion disk are split into two parts: the calculation of the transfer function and the calculation of the local spectrum at any emission point in the disk. The transfer function only depends on the background metric and takes into account all the relativistic effects (gravitational redshift, Doppler boosting, and light bending). Our code computes the transfer function for a spacetime described by the Johannsen metric and can easily be extended to any stationary, axisymmetric, and asymptotically flat spacetime. Transfer functions and single line shapes in the Kerr metric are compared to those calculated from existing codes to check that we reach the necessary accuracy. We also simulate some observations with NuSTAR and LAD/eXTP and fit the data with our new model to show the potential capabilities of current and future observations to constrain possible deviations from the Kerr metric.