I am confusing whether I could use "convenient" and "any" at the same time.

Also, I'd like to know the different nuance between may and can, in this sentence:

You may call me in your convenient any time.


You can use "any" and "convenient" in the same sentence, but not the way you wrote it.

You can tell someone they can call you whenever they want by saying "at any time" (sometimes this is shortened to "anytime"):

You can call me at any time.
You can call me anytime.

Or, you can emphasize that they can call at a time when it would be convenient for them:

You can call me whenever it's convenient for you.

The phrase "at your convenience" is sometimes used:

You can call me at your convenience.

If I were to combine the two, I'd probably put the "any time" part first:

You can call me anytime that's convenient for you.

If the part about convenience goes first, then anytime can be used for emphasis:

You can call me at your convenience – anytime, day or night.

As for may vs. can, may is probably the more correct word to use (at least in a more formal sense), but can might sound more friendly in informal contexts. But you could say the sentence without using either one:

Call me any time it's convenient.

Or, if you really want to be brief, this works, too:

Call me anytime!

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