Obtaining strong magnetoelectric couplings in bulk materials and heterostructures is an ongoing challenge.We demonstrate that manganite heterostructures of the form (Insulator)/(LaMnO3)n/Interface/(CaMnO3)n/(Insulator) show strong multiferroicity in magnetic manganites where ferroelectric polarization is realized by charges leaking from LaMnO3 to CaMnO3 due to repulsion. Here, an effective nearest-neighbor electron-electron (electron-hole) repulsion (attraction) is generated by cooperative electron-phonon interaction. Double exchange, when a particle virtually hops to its unoccupied neighboring site and back, produces magnetic polarons that polarize antiferromagnetic regions. Thus a striking giant magnetoelectric effect ensues when an external electrical field enhances the electron leakage across the interface.