Recent conceptual work has shifted from the individual to the group by positing compassion as a collective capacity that involves noticing others' suffering, feeling empathic concern, and attempting to alleviate that suffering.
The answer is definitively yes, compassion is both innate and can be learned and enhanced.
Advances in neuroscience have shown us that the human brain has neural networks that are hard-wired with the ability to share the experiences of others, including emotions and sensations.
This consistency strongly suggests that compassion isn't simply a fickle or irrational emotion, but rather an innate human response embedded into the folds of our brains.