Right now there are two big things for biology on that front.
1) Straight up clones
2) Splicing things in
Cloning humans is banned in many places, mainly as most clones have a lot of issues (advanced ageing for one). Should those be solved adequately (risks brought into line with general reproduction sort of thing) I imagine that would go away fairly quickly.
Inserting the human genome into things on the other hand raises heaps of ethical issues, mainly as the rights of the resulting creature would have to be determined -- there are some intelligent animals out there in the world but there are some massive gaps between humans and them. You create a halfway point and that might not be cool. Anything on that front will likely only happen when our understanding of genetics is considerably more advanced, or the techniques become far easier to do in lesser funded situations.
If you mean geep (sheep + goat) style hybrid where embryos are combined and you end up with some parts one animal, some parts another then I can see that happening far sooner. Indeed research into producing sperm of one animal in mice, humans included in this though I am not sure to what extent, has been going on for well over a decade now.