Efficient coupling of soft robotic cardiac assist devices to the external surface of the heart is crucial to augment cardiac function and represents a hurdle to translation of this technology. In this work, we compare various fixation strategies for local and global coupling of a direct cardiac compression sleeve to the heart. For basal fixation, we find that a sutured Velcro band adheres the strongest to the epicardium. Next, we demonstrate that a mesh-based sleeve coupled to the myocardium improves function in an acute porcine heart failure model. Then, we analyze the biological integration of global interface material candidates (medical mesh and silicone) in a healthy and infarcted murine model and show that a mesh interface yields superior mechanical coupling via pull-off force, histology, and microcomputed tomography. These results can inform the design of a therapeutic approach where a mesh-based soft robotic DCC is implanted, allowed to biologically integrate with the epicardium, and actuated for active assistance at a later timepoint. This strategy may result in more efficient coupling of extracardiac sleeves to heart tissue, and lead to increased augmentation of heart function in end-stage heart failure patients.