Variable selection is a crucial issue in model building and it has received considerable attention in the literature of survival analysis. However, available approaches in this direction have mainly focused on time-to-event data with right censoring. Moreover, a majority of existing variable selection procedures for survival models are developed in a frequentist framework. In this article, we consider additive hazards models in the presence of current status data. We propose a Bayesian adaptive least absolute shrinkage and selection operator (lasso) procedure to conduct a simultaneous variable selection and parameter estimation. Efficient Markov chain Monte Carlo (MCMC) methods are developed to implement posterior sampling and inference. The empirical performance of the proposed method is demonstrated by simulation studies. An application to a study on the risk factors of heart failure disease for type 2 diabetes patients is presented.