No particular time, as in It doesn't matter when; any day is fine with me.
Also, any day now. Quite soon, as in I might get a call any day, or There could be a snowstorm any day now.
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.
Examples from the Web for any day
Historical Examples of any day
He also seems to think lynching an any-day possibility in the streets of New York.The Land of Contrasts
James Fullarton Muirhead