To begin with, I think it’s easy to make category errors in confusing the metaphorical and the physical heart. The heart sounds warm and fuzzy while the brain sounds gushy or “head-y.” So it’s tempting to ascribe heartfelt feelings/inclinations to the physical heart even though it’s actually the brain that (still somewhat mysteriously) constructs those feelings and inclinations as experiences in our field of awareness.
As to the physical heart and the brain, there is interesting research on heart rate variability from Dacher Keltner, Heartmath, and other sources. I think sometimes there is an overemphasis on the heart’s influence on the brain and thus our experience of things like warm and loving feelings. Still, clearly there are feedback loops in which cardio-pulmonary processes regulate neural and thus mental activity. In effect, through Heartmath or related methods (e.g., metta practice in Buddhism, devotional practices in Hinduism), one can use the mind to affect the brain to affect the physical heart, which in turn affects the brain, which causes the mind to feel more peaceful/loving/happy. I have not seen research to this effect, but it is certainly plausible that changes in a person’s mental state – such as becoming more loving – involve functional and structural changes in the neuro-fabric of the heart.
Polyvagal theory is also a tremendous resource that indicates linkages between the heart and lungs and the “social engagement system.”