When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study the possible behaviors of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the infinite-dimensional inverse problem - the underlying motivations being the stability or instability of infinite-dimensional Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an infinite-dimensional inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem.