Can this mean (ignoring likelihood or awkwardness) both
He says I'm eating
He starts talking if/when I'm eating
I ignored likelihood and awkwardness. I assumed that you are only interested in usage of と and 話す, and that it is allowed to add any contexts.
「食べていると話す」can both mean "X tells Y that Z is eating" or "X talks to Y while Z is eating."
However, the pitch-accents are different in standard Japanese.
「食べているとはなす」(X tells Y that Z is eating.)
「食べているとはなす」(X talks to Y while Z is eating.)
(Bold means high pitch accent, otherwise low. There were six native speakers around me when I noticed this, and five of them agreed with me. One said no difference. Yet, the one is not from Kanto area.)
I must agree that there is awkwardness in the former sentence when this has no context. This comes from the grammatical aspects and the meanings.
「話す」has no aspect nor tense (it is not 話している or 話した for example.) So, it might be habitual or something in quite general situation. And,「話す」is "to talk" or "to tell", which "needs to relay some information." However, "食べている (continuous aspect)" itself rarely has much information, yet there could be a special context that whether "食べている" or not is really important. These facts make the situation awkward. You need to come up a special context that "食べている" almost always has much information.
Well, I think the latter sentence is also strange. It is hard to imagine a person who only talks in a certain condition.
So, if you change「話す」to continuous aspect and change「食べる」, you can make more natural sentence easier.
The arrested terrorist is telling the government that their group has caught the missing president.
I tried to make a natural sentence out of「食べていると話す」, which was hard.
Maudgalyayana (one of the Buddha's closest disciples) can see the hell, and he sometimes tells that Oni is eating a dead.
I googled and found about 50 results, all of them were "X tells Y that Z is eating" type.
Example from web:
Sasaki tells that she usually eats various dishes with PIETORO sauce.
Slim women tend to say that "I do not diet" or "I eat a lot."
These two are interesting. The former is an advertisement of sauce and the latter is an article about a diet so that eating itself is important.
Lastly, I could not find "X talks to Y while Z is eating." type sentences, so let me make up one.
The old talks about the war while I am eating carrots.