Make sure said villager is at a high friendship level.
Try and find a mover by speaking to animals. If someone is thinking about moving, stop them if you dont, let them if you do. If you dont, save and quit, go to system setting and go forward by 3 days. Play game again, and someone else will most likely think about moving. If you find no mover, tt 1 day ahead.
Its my personal way, and its worked for me.