The optimal functioning of nanopore-based biosensing tools necessitates rapid polymer capture from the ion reservoir. We identify an ionic correlation-induced transport mechanism that provides this condition without the chemical modification of the polymer or the pore surface. In the typical experimental configuration where a negatively charged silicon-based pore confines a 1:1 electrolyte solution, anionic polymer capture is limited by electrostatic polymer-membrane repulsion and the electroosmotic (EO) flow. Added multivalent cations suppress the electrostatic barrier and reverse the pore charge, inverting the direction of the EO flow that drags the polymer to the trans side. This inverted EO flow can be used to speed up polymer capture from the reservoir and to transport weakly or non-uniformly charged polymers that cannot be controlled by electrophoresis.