The recently proposed function controlled variable step-size least-mean-square (FCVSSLMS) algorithm has shown high performance in different noise environments. The performance of the algorithm can be improved further if the system is sparse. In this paper, we propose a new algorithm based on the FCVSSLMS algorithm. The proposed algorithm imposes an approximate penalty in the cost function of the FCVSSLMS algorithm. We also present the convergence analysis of the proposed algorithm and derive the stability criterion. The performance of the proposed algorithm is compared to those of the variable step-size least-mean-square, more robust variable step-size least-mean-square, FCVSSLMS algorithm and reweighted zero-attracting least-mean-square algorithm in a system identification setting with an additive white Gaussian noise and additive correlated Gaussian noise. The proposed algorithm has shown high performance compared to the others in terms of the convergence rate and mean square deviation.