Which of the following options is the correct or the acceptable?
"He is an English native speaker"
"He is a native English speaker".
I can't see any good reason for supposing that either version is "correct" (or by implication, that the other is "incorrect"), but there's certainly a big difference in prevalence...
I can't see any way to justify the possibility that one sequence might actually mean something different to the other1. They seem to net down to the same thing to me...
1 EDIT: Thanks to comments from @ErikE and @psmears below, I must admit that once it's pointed out to me, I can naturally parse an English native speaker as someone from England. But unless context strongly forced a perverse interpretation, I'd take it for granted that meant an Anglophone, since there's no such thing as an English passport or an officially "naturalized" Englishman. To be English you really have to be born and brought up in England, which pretty much dictates that English will be your mother tongue.
I think the first sentence involves a lexical ambiguity due to the fact that the word "English" can mean (at least) two very different things:
This is ambiguous. It could mean "He is a person who speaks the English language natively," or it could mean "He is an English person who speaks an unspecified language natively."
Here is an illustrative related sentence:
This sentence happens to describe me; I am not English, but I am a native speaker (of some unspecified language). The lexical ambiguity is avoided in the above American part because there is no American language.
This provides no ambiguity. "English speaker" can only refer to the language, and not the nationality. (Now technically, I suppose this could also mean "He is a native, who is from England, who speaks about unspecified things from time to time," but using "English speaker" in such a way is quite odd. I think that kind of possible meaning would only occur to an AI attempting to parse natural language, or something strange like that.)
I think the ambiguity in the first usage is why @FumbleFingers ' n-gram plot favors this latter construction. There are many Scottish native English speakers, and Indian native English speakers, etc.
As others have posted, the correct answer is a native English speaker, not the inversion an English native speaker.
This is because you're not listing two adjectives that describe speaker, you're actually describing an English speaker who is also native.
Others have brought up the idea of an English/English(man) distinction. I would have to encounter a very twisted sentence in order to think of this noun phrase ever meaning a person from England who is a native speaker of a language that is not English.
I would probably rephrase it to something like "English is his primary language". In a different context, we could use "can you find someone who speaks English as a primary language?" It's somewhat longer than "can you find a native English speaker?" but avoids the ambiguity.
I really don't think anyone would misunderstand either of your examples unless they were being deliberately obtuse, but neither seems very natural to me, despite English being my primary language. I can just imagine a case where I'm in France looking for a native who speaks any English at all, and someone thinks I'm looking for someone who primarily speaks English. But it seems likely the context would make my intent clear even in a contrived situation like this.
Lets see... how to remember this...
1) What are you looking for?
What about the speaker of English?
search results will give you native English speaker
2) What are you looking for?
search results will lead you to this interesting article:
But to continue, What kind of English speaker?
search results will get you back to native speaker of English
3) What are you looking for?
search results will get you a definition for native speaker which would indicate that English native speaker is someone who is English and a native speaker of something.
So it would seem native English speaker is more useful than English native speaker since in order of importance being English speaking is more important than being a native speaker of any language which is probably the case because you're interested in English, and then within the group of English speakers the native ones are the ones you are interested in
By pure definition, a native English speaker could be someone who grew up in a nonEnglish speaking country but, for example, went to English speaking schools and had English speaking parents and primarily used English to communicate (think: foreign posted military family)
This is a question about which compound noun is better when combined together between native speaker and English speaker.
People are all native speakers of one specific language (sometimes two depending on the country where they live), but English speaker is more specific than native speaker as it means only those who speak English.
Therefore, it is more idiomatic for the adjective native to modify English speaker than the other way around. The linked Ngram Viewer clearly favors native English speaker.
It doesn't necessarily mean that "English native speaker" wouldn't make any sense. It makes sense, but less idiomatic than "native English speaker".