A rigorous criterion for a finite system of plane vortices to be close to the set of all collapsing configurations is given. The criterion is a consequence of a general theorem (based on the topological degree of a map) on the existence of solutions of the nonlinear systems of equations, combined with some analytical bounds for derivatives of the equations satisfied by collapsing configurations. The criterion can be used to prove the existence of collapsing configurations of vortices for many circulations for which the existence theory of O'Neil does not work. A detailed application of the criterion to a sample collapsing system of seven vortices is given.