How about almost fluent?

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Note added at 9 mins (2004-08-04 15:14:42 GMT)
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Well, google search for \"near fluent\" got 1540 hits, \"almost fluent\" got 1650 hits and \"virtually fluent\" got 100 hits.
So, you can use either \"near fluent\" or \"almost fluent.\"

Another possibility.

Monterey Institute of International Studies
Instructors:
All instructors have native or near-native fluency and experience with the methodology of intensive language learning.
http://www.miis.edu/lang-ilp-langs-russian.html

MIGHT be the best answer. To qualify fluency, which means the ability to speak (and be understood!) and understand other speakers in a language not your own rather tends to devalue your "fluency". I do agree there are grades of fluency but I feel that in a CV the expressions "outstanding/excellent/very good/ good" might go down better.

I would say that grammatically you should say "nearly fluent" rather than "near fluent". On the whole, as the concept of "nearly fluent" is somewhat subjective, I would prefer something like: Proficiency level English. passed with distinction. This would be an equivalent to "nearly fluent".
Another option in use in Brazil is "native-like" but I would not use it here, as I take this to mean someone with native fluency but not actually a native speaker.

Why not just say that you speak good French (or whatever) - or excellent French, if you want to impress even more. To me, fluent is a dodgy term to use - as others have said, it is hard to tell exactly what it means. Is fluent supposed to mean that you can barely be distinguished from a native speaker (that would be the near-native fluency that has been proposed), or simply that you can speak it well enough to manage fine for all practical purposes? If one takes the second interpretation (as some people will if you use the term), then "almost fluent" could be interpreted as somewhat disparaging and not particularly good. Rather than say something with slightly negative connotations ("not good enough to be fluent"), it would be better to say something entirely positive, which is why I suggest good or excellent.