Retinal vein cannulation is a promising approach for treating retinal vein occlusion that involves injecting medicine into the occluded vessel to dissolve the clot. The approach remains largely unexploited clinically due to surgeon limitations in detecting interaction forces between surgical tools and retinal tissue. In this paper, a dual force constraint controller for robot-assisted retinal surgery was presented to keep the tool-to-vessel forces and tool-to-sclera forces below prescribed thresholds. A cannulation tool and forceps with dual force-sensing capability were developed and used to measure force information fed into the robot controller, which was implemented on existing Steady Hand Eye Robot platforms. The robotic system facilitates retinal vein cannulation by allowing a user to grasp the target vessel with the forceps and then enter the vessel with the cannula. The system was evaluated on an eye phantom. The results showed that, while the eyeball was subjected to rotational disturbances, the proposed controller actuates the robotic manipulators to maintain the average tool-to-vessel force at 10.9 mN and 13.1 mN and the average tool-to-sclera force at 38.1 mN and 41.2 mN for the cannula and the forcpes, respectively. Such small tool-to-tissue forces are acceptable to avoid retinal tissue injury. Additionally, two clinicians participated in a preliminary user study of the bimanual cannulation demonstrating that the operation time and tool-to-tissue forces are significantly decreased when using the bimanual robotic system as compared to freehand performance.