While linear regression focuses on studying the relationship between a set of covariates and a real-valued response, Fréchet regression addresses more complex responses, with the sole assumption that observations can be compared using a metric. Examples of such metric space-valued responses include correlation matrices, distributions, graphs, trees, spherical data, and many others. In this work, we propose a Bayesian approach to Fréchet regression, allowing practitioners to incorporate prior knowledge into the regression process. One of the main challenges is that Fréchet regression does not directly describe a model nor does it involve parameters, making prior specification non-trivial. By drawing inspiration from Bayesian linear regression and importance weighting, we can overcome these challenges and provide a practical and tractable solution.