Einstein-Podolsky-Rosen (EPR) steering, an important resource in quantum information, describes the ability of one party to influence the state of another party through local measurements. It differs from Bell nonlocality and entanglement due to its asymmetric property. EPR steering swapping allows two spatially independent parties to present EPR steering without direct interaction. Here, we theoretically propose an all-optical EPR steering swapping (AOSS) scheme based on low-noise high-broadband optical parametric amplifiers (OPAs). A low-noise high-broadband OPA is utilized to implement the function of Bell-state measurement without detection, avoiding the optic-electro and electro-optic conversions. After AOSS, one-way and two-way EPR steering between two independent optical modes can be obtained. Our scheme provides a guidance for the construction of a measurement-free, all-optical broadband quantum network utilizing EPR steering swapping.