If I were to ask a friend to call me at his convenience; how am I supposed to say it? Is it correct to say 'Do make a call'?
I don't think anyone speaks that way to friends. I hear: "Call me," or "(you should) give me a call."
[nat]
1: Hey, when should I call you?
2: Just call me at your convenience.
Sounds perfectly fine to me. I've certainly said that to friends.