  Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in
\mathbb{F}$. In this article we establish a sufficient condition for the
existence of a primitive normal pair $(\alpha, f(\alpha)) \in \mathbb{F}_{q^m}$
over $\mathbb{F}$ with a prescribed primitive norm $a$ and a non-zero trace $b$
over $\mathbb{F}$ of $\alpha$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a
rational function of degree sum $n$ with some minor restrictions. Furthermore,
for $q=7^k$, $m \geq 7$ and rational functions with numerator and denominator
being linear, we explicitly find at most 6 fields in which the desired pair may
not exist.
