The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and indeed the Korovkin-type approach, in the setting of preconditioning large linear systems with Toeplitz structure, is well known. Here we extend this finite-dimensional approach to the infinite-dimensional context of operators acting on separable Hilbert spaces. The asymptotics of these preconditioners are evaluated and analyzed using the concept of completely positive maps. It is observed that any two limit points, under Kadison's BW-topology, of the same sequence of preconditioners are equal modulo compact operators. Moreover, this indicates the role of preconditioners in the spectral approximation of bounded self-adjoint operators.