As a classical sufficient dimension reduction method, sliced inverse regression (SIR) (Li, 1991) replaces the original predictors by their low-dimensional linear combinations while preserving all the relevant information between the response and the predictors. After partitioning the response into different slices, the intraslice means of the predictors are used to recover such linear combinations. However, these linear combinations involve all the predictor variables. To address this limitation, we propose two Bayesian model averaging (BMA) approaches to achieve sparse sufficient dimension reduction. While the first approach leverages univariate response BMA (Raftery et al., 1997) and deals with each slice separately, our second approach uses multivariate response BMA (Brown et al., 1998) to estimate the intraslice means jointly. Through extensive simulation studies, the new proposals are shown to outperform SIR and existing sparse sufficient dimension reduction methods such as sparse SIR (Li, 2007), coordinate-independent sparse estimation (Chen et al., 2010), and frequentist model averaging SIR (Fang and Yu, 2020).