Both of these spellings appear to be correct - and in common use. At least, that's what my internet search turned up. But which is considered the most correct form?
In this native speaker's opinion, "eyeing" looks much better. "Eying" may be correct according to dictionarieis, but it's such an odd-looking word it would take me a minute to figure out what it was.
According to two dictionaries, both are correct. The Cambridge Learner's Dictionary says that eying is American.
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I am a native speaker. "Eyeing" is more pleasing to me. "Eying" looks weird.