Villagers will never leave without checking in with you first.
However; they CAN want to leave (have a thought bubble asking you if it's okay) on Saturdays, Sunday or during special events too.
I had Sherb wanting to move during an event.
i've never had someone prompt to move during a saturday or an event, but i get sundays and rainy days all the time lol. my guess is, pure RNG makes anything possible!
They don't ask to move on Saturdays if you have K.K. but if for some reason, he hasn't been invited yet (like you aren't 3 star yet), they could.. They could ask to move on Sundays though, based from experience.
I've never had a villager ask to leave on a Saturday or a Sunday, but that might be just because the game is so new that I've only experienced a small handful of move-out cycles. If you miss playing on a day where a villager does want to move, the worst thing that'll happen is that the game will pick a different villager to want to move out the next time you load up the game.
Did you buy on launch and haven't TT'd/do the lil trick where you quit the game and ignore a villager who plans to move out who you dont want to move out? Because same, and I'm expecting a move out tomorrow!
I'll also feedback. A video I watched said that it's more improbable on Saturdays, but it's still possible!