Let and be a smooth generic representation of (split) and over a non-archimedean field respectively with . Suppose that is irreducible and has finite length that admits a unique Whittaker model. Then Gross and Tsai have proposed a conjectural new and old forms theory for , which puts results of Casselman and Roberts-Schmidt into a single framework. On the other hand, Soudry has developed a local theory for Rankin-Selberg convolutions for generic representations of , which represent the standard tensor product -factors. In this talk, we consider the Rankin-Selberg integrals for attached to (conjecutral) new and old forms in . We show that such integrals always vanish if is ramified and compute the integrals when is unramified.