We address the problem of Bayesian variable selection for high-dimensional lin-ear regression. We consider a generative model that uses a spike-and-slab-like prior distribution obtained by multiplying a deterministic binary vector, which traduces the sparsity of the problem, with a random Gaussian parameter vector. The origi-nality of the work is to consider inference through relaxing the model and using a type-II log-likelihood maximization based on an EM algorithm. Model selection is performed afterwards relying on Occam's razor and on a path of models found by the EM algorithm. Numerical comparisons between our method, called spinyReg, and state-of-the-art high-dimensional variable selection algorithms (such as lasso, adap-tive lasso, stability selection or spike-and-slab procedures) are reported. Competitive variable selection results and predictive performances are achieved on both simulated and real benchmark data sets. An original regression data set involving the predic-tion of the number of visitors of the Orsay museum in Paris using bike-sharing system data is also introduced, illustrating the efficiency of the proposed approach. An R package implementing the spinyReg method is currently under development and is available at https://r-forge.r-project.org/projects/spinyreg.