This phrase is not always as polite as people believe.
The phrase at my earliest convenience shows up frequently in business communications, often in a way that is less courteous than the user thinks it is.
For example, the phrase can sometimes be heard at the end of outgoing voicemail greetings, as in: “This is Mary Jones. I’m sorry I missed your call. Please leave a message, and I will call you back at my earliest convenience.”
Mary may think the phrase demonstrates her responsiveness, but let’s say a client calls Mary and gets her voicemail. Even though Mary is offering to call the client back at her earliest convenience, she is still essentially saying she will not call back before it becomes convenient for her. And wouldn’t the ideal of good customer service be to base the timing of the return call on the convenience of the client? It may well be convenient for the client to hear back from Mary long before it actually feels convenient to Mary to make that call.
Better, then, for Mary to end her greeting with something along the lines of “I will call you back as soon as possible.”