The estimation of the polarization $P$ of extragalactic compact sources in Cosmic Microwave Background images is a very important task in order to clean these images for cosmological purposes -- as, for example, to constrain the tensor-to-scalar ratio of primordial fluctuations during inflation -- and also to obtain relevant astrophysical information about the compact sources themselves in a frequency range, $ u sim 10$--$200$ GHz, where observations have only very recently started to be available. In this paper we propose a Bayesian maximum a posteriori (MAP) approach estimation scheme which incorporates prior information about the distribution of the polarization fraction of extragalactic compact sources between 1 and 100 GHz. We apply this Bayesian scheme to white noise simulations and to more realistic simulations that include CMB intensity, Galactic foregrounds and instrumental noise with the characteristics of the QUIJOTE experiment Wide Survey at 11 GHz. Using these simulations, we also compare our Bayesian method with the frequentist Filtered Fusion method that has been already used in WMAP data and in the emph{Planck} mission. We find that the Bayesian method allows us to decrease the threshold for a feasible estimation of $P$ to levels below $sim 100$ mJy (as compared to $sim 500$ mJy that was the equivalent threshold for the frequentist Filtered Fusion). We compare the bias introduced by the Bayesian method and find it to be small in absolute terms. Finally, we test the robustness of the Bayesian estimator against uncertainties in the prior and in the flux density of the sources. We find that the Bayesian estimator is robust against moderate changes in the parameters of the prior and almost insensitive to realistic errors in the estimated photometry of the sources.
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