This paper studies a new Bayesian algorithm for fusing hyperspectral and multispectral images. The observed images are related to the high spatial resolution hyperspectral image to be recovered through physical degradations, e.g., spatial and spectral blurring and/or sub-sampling defined by the sensor characteristics. In this work, we assume that the spectral response of the multispectral sensor is unknown as it may not be available in practical applications. The resulting fusion problem is formulated within a Bayesian estimation framework, which is very convenient to model the uncertainty regarding the multispectral sensor characteristics and the scene to be estimated. The high spatial resolution hyperspectral image is then inferred from its posterior distribution. More precisely, to compute the Bayesian estimators associated with this posterior, a Markov chain Monte Carlo algorithm is proposed to generate samples asymptotically distributed according to the distribution of interest. Simulation results demonstrate the efficiency of the proposed fusion method when compared with several state-of-the-art fusion techniques.