AbstractWe construct a relativistic reflection model in a non-Kerr spacetime in which, depending on the value of the deformation parameter of the metric, there are black hole solutions with spin parameter $|a_*| > 1$|a∗|>1. We apply our model to fit Suzaku data of four Seyfert galaxies (Ton S180, Ark 120, 1H0419–577, and Swift J0501.9–3239). These galaxies host at the center supermassive black holes that were previously interpreted as near-extremal Kerr black holes. For Ton S180 and 1H0419–577, our measurements are still consistent with the Kerr hypothesis. For Ark 120 and Swift J0501.9–3239, the Kerr solution is not recovered at 3-$σ $σ. We discuss our results and possible systematic uncertainties in the model.