Acoustic reflector estimation, which is one of the key problems of robot audition, is addressed in this paper using a sparse Bayesian learning (SBL) approach. More specifically, we propose a three-step procedure in which we 1) reconstruct the room impulse response (RIR) using SBL, 2) estimate the time-of-arrivals (TOAs) from the RIR, and 3) estimate the direction of arrival (DOA) from the TOA estimates. The challenge of RIR reconstruction is that the early reflections are weak compared to the direct sound. Therefore, the sparsity of the early part of the RIR is exploited to improve the recovery performance. However, most of the sparse vector recovery methods can not reconstruct the RIR successfully, especially when the measurement matrix is highly coherent. In this paper, we therefore adopt the SBL framework which is more robust in such scenarios compared to state-of-the-art recovery methods. In the DOA estimation step, we propose a new approximate near-field model for isotropic arrays. The performance of the proposed approach is analysed by numerical simulations, where the estimation accuracy is measured versus different signal-to-diffuse-noise ratios and grid errors. According to the simulation results, the proposed SBL method is more robust to diffuse noise and grid errors than other state-of-the-art methods that even fails to estimate the TOAs and DOA in many cases.