I am sure the method I am going to describe has been posted a few times all over the internet, but here’s what I do which usually takes me around 1 1/2 to 2 hours to actually get the one to leave which I want:
1. On the current day talk to every villager, then hit the home button to go to your consoles time and date
2. Move the console time forward to around 15 days in the future, around 1 pm has usually worked for me the best.
3. Go back to the game save&end, then start back up immediately
4. Look for the thought bubble
5. If there is no thought bubble proceed forward another day
6. If there is a thought bubble:
a)It’s the villager you want gone: Say bye, save & exit and proceed on to the next day to get them into boxes and so on.
b)you want to keep the villager: Leave the game via Home Button and close the software without saving. Note the date and time the bubble appeared and tt back to the starting (current)date.
6. Proceed on from step 1) and tt to the date and time you had the first thought bubble appear.
Notes: Usually there will be some other villager asking to leave, however sometimes you still need to tt forward another day if no one shows up.
Additionally I had incidents where I had the thought bubble appear on the current date of villagers I didn’t want to leave, just tell them “don’t go” and proceed on with the tt.
This was written on mobile, so sorry if my formatting is **** lol.