We present a novel methodology to perform Bayesian variable selection in finite mixture model of linear regressions, particularly in the presence of heavy-tailed distributions. The new method considers the observations come from a heterogeneous population which is a mixture of a finite number of sub-populations. Within each sub-population, the response variable can be explained by a linear regression on the predictor variables. Moreover, we explore to identify different subsets of variables that are correlated to the response in each sub-population and are robust to outliers in the data. Inference is performed via Markov chain Monte Carlo---a Gibbs sampler with Metropolis-Hastings steps for a class of parameters. Simulated studies highlight the performance of this approach when covariates are highly correlated with various selection criteria. Examples with exchange market pressures during the recent global financial crisis are presented and an extension to mixture models with an unknown number of components is introduced and discussed.