Multi-organ perfusion systems offer the unique opportunity to mimic different physiological systemic interactions. However, existing multi-organ culture platforms have limited flexibility in specifying the culture conditions, device architectures, and fluidic connectivity simultaneously. Here, we report a modular microfluidic platform that addresses this limitation by enabling easy conversion of existing microfluidic devices into tissue and fluid control modules with self-aligning magnetic interconnects. This enables a 'stick-n-play' approach to assemble planar perfusion circuits that are amenable to both bioimaging-based and analytical measurements. A myriad of tissue culture and flow control TILE modules were successfully constructed with backward compatibility. Finally, we demonstrate applications in constructing recirculating multi-organ systems to emulate liver-mediated bioactivation of nutraceuticals and prodrugs to modulate their therapeutic efficacies in the context of atherosclerosis and cancer. This platform greatly facilitates the integration of existing organs-on-chip models to provide an intuitive and flexible way for users to configure different multi-organ perfusion systems.