I'm working with Spring + JPA (with hibernate as JPA provider) using services and dao.
I'd like to inject the JPA
You can inject EntityManager using @PersistenceContext which will inject shared EntityManager managed by Spring. But i think you should reconsider your approach of having EM in service class as well.
Also look here : Doc
As OP's comments updating : you can have EntityManager only in DAO classes but make those to-be atominc dao calls in same service method and make it transactional.Check out Two Dao atomic calls