I am texting a friend who might be busy. So I text him ' leave a message when you are free'? Is that correct?
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1Both of them are correct. – user61367 Sep 16 '17 at 18:44
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Why not just text the question? Everybody knows there will be a response when the recipient is free, otherwise not (depending on how urgent the message seems). – Weather Vane Sep 16 '17 at 20:23
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Using become in that context borders on suggesting that your friend is incarcerated in some way. – Lawrence Sep 17 '17 at 6:00
The scenario you present is so common that nuance plays very little, if any, role in it.
But let's say that you wanted to express the idea that you respect the other person's time; it is valuable and in great demand. In that case, "when you become free" or "when you have a free moment" might do. The phrase "when you're free" is a little too casual to express anything more than that you realize there may be some delay in the response.
Both your examples
When you are free
When you become free
are essentially the same, obviously your friend will be free when he becomes free.
Since it is a friend, you might use
When you have a moment
When you get a chance
whereas
When it is convenient
might be considered more polite/formal.