No set or specific time or date; any time. I don't need the report back immediately, any day will do.
1. No particular time, as in It doesn't matter when; any day is fine with me.
2. Also, any day now. Quite soon, as in I might get a call any day, or There could be a snowstorm any day now.
3. Also, any day of the week. Every day, as in I could eat fresh corn any day of the week. All three senses employ any in the sense of "no matter which," a usage dating from a.d. 1000.
any day1 at any time. 2 used to express your strong preference for something under any circumstances.
2 2003 Royal Academy Magazine Give me Bruegel rather than Michelangelo any day.