In this paper, we propose sparsity-aware data-selective adaptive filtering algorithms with adjustable penalties. Prior work incorporates a penalty function into the cost function used in the optimization that originates the algorithms to improve their performance by exploiting sparsity. However, the strength of the penalty function is controlled by a scalar that is often a fixed parameter. In contrast to prior work, we develop a framework to derive algorithms that automatically adjust the penalty function parameter and the step size to achieve a better performance. Simulations for a system identification application show that the proposed algorithms outperform in convergence speed existing sparsity-aware algorithms.