The phrase "at any time convenient for you" is correct and usable in written English. You can use it when you need to make sure that the person or people you are speaking to are able to do something at a time that suits them. For example, you could say: "Let me know when you would like to meet, at any time convenient for you.".
At the news conference today, Mr. Putin extended an invitation for Mr. Bush to visit Russia "in any format, at any time convenient".
I turn up at a department at a time convenient for me, work only for my allotted time and, a week later, am paid.
For example, some well women chose to seek care at a time "convenient" for them.
Respondents also have more flexibility to complete the survey at a time convenient for them [ 36].
Trained local enumerators visited the respondents' households and conducted 60-min interviews at a time convenient for each respondent.
The follow-up interview occurred at a time convenient for the participant, up to one year after delivery.
The telephone interviews occurred two to four weeks after the observation at a time convenient for the woman.
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Ha Thuy Vy
MA of Applied Linguistic, Maquarie University, Australia