  In this paper, we consider rational functions $f$ with some minor
restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some
prime $p$ and positive integer $k$. We establish a sufficient condition for the
existence of a pair $(\alpha,f(\alpha))$ of primitive normal elements in
$\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}.$ Moreover, for $q=2^k$ and rational
functions $f$ with quadratic numerators and denominators, we explicitly find
that there are at most $55$ finite fields $\mathbb{F}_{q^n}$ in which such a
pair $(\alpha,f(\alpha))$ of primitive normal elements may not exist.
