Iteratively reweighted least squares (IRLS) method is a popular approach for sparse signal recovery, provably achieves superior performance in a series of applications. Nevertheless, iteratively reweighted least squares involves a matrix inversion at each iteration, which makes it impractical in massive data. To address this issue, we introduced approximate inverse-free accelerated IRLS methods in this paper. Specifically, by adopting the approximate inverse preconditioning with conjugate gradient, we obtain an approximate inverse-free IRLS method. Moreover, a nesterov acceleration scheme and decreased parameters with fixed scale strategy are independently adopted to accelerate the approximate inverse-free IRLS method, which leads to two computationally efficient approximate inverse-free accelerated IRLS algorithms. Experimental results clearly demonstrate the superiority of our proposed approaches, which not only achieve a comparable sample complexity but also have faster convergence with less execution time when compared to traditional fast algorithms.