Big 2 you can play with a mix of bots and human players. In a manual table setup, you just have to hit "Invite Player..." and bots should be in that list if they are connected.
Hearts I haven't programmed any bots for. I'll keep it in mind though. Currently working on a newer version of this site with better mobile support (though suspect it will take a while until I can get anything released as I still have my "regular" job).
Will there ever be bots in Hearts? Also: Is there a way to play with one human opponent and two bots in Hearts? If not, could that be constructed? I have the same question for Big Two.