In many applications of wireless sensor networks (WSNs), sensor positions are often not known exactly. The existence of sensor position error (SPE) may significantly impact system performance if not appropriately modeled or considered. In this article, we address the important sensor selection problem in the presence of SPE for general nonlinear measurement models considering independent and correlated measurement noise cases. The sensor selection problem is formulated in the framework of sparse sensing, in which the number of activated sensors is minimized subject to certain predetermined performance constraints. To facilitate the challenging convex relaxation of nonconvex constraints for the case with independent measurement noise, we prove that the Fisher information matrix (FIM) remains additive even in the presence of SPE. For correlated measurement noise, quadratic inequality constraints are introduced and two suboptimal solvers are proposed. The first solver uses matrix decomposition to transform the quadratic constraint into linear matrix inequalities (LMIs), while the second solver iteratively performs a linearization procedure on the quadratic constraint to obtain a reduced dimension of LMI, thus decreasing the computational complexity significantly. The proposed algorithms are compared in terms of their computational complexity quantitatively and experimentally with suboptimal greedy approaches and existing algorithms ignoring SPE. The results show the importance and necessity of considering SPE when implementing sensor selection and demonstrate the effectiveness of the proposed three sensor selection solvers.