If you have chest pain when swallowing, could this be the heart or the stomach?

Probably esophagus: Pain when swallowing could be an esophageal problem. It could be simple dysmotility, reflux, to more serious cancer. Don't fool around with this one.

Both: It could be atypical angina which is pain from heart disease. Or it could be peptic ulcer disease which involves the stomach. But it is more likely to derive from the esophagus.