  Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$
be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we
establish a sufficient condition for the existence of primitive normal pairs of
the type $(\epsilon, f(\epsilon))$ in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$
with two prescribed traces,
$Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b$, where $f(x) \in
\mathbb{F}_{q^m}(x)$ is a rational function with some restrictions and $a, b
\in \mathbb{F}_q$. Furthermore, for $q=5^k$, $m \geq 9$ and rational functions
with degree sum 4, we explicitly find at most 12 fields in which the desired
pair may not exist.
