Does whenever convenient sound natural to you in the following sentence?
Please call my secretary to arrange a meeting this afternoon, or whenever convenient to you.
Does the sentence sound natural to you?
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to is your main problem. Something isn't convenient to you, it's convenient for you. Compare:
"whenever it is convenient for you" would be the textbook answer.
Still, it's understandable contracted as
"whenever convenient for you." It would even be understood if it was contracted to "Please call my secretary to arrange a meeting this afternoon, or whenever convenient."