  Given $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$,
$a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of
primitive pair $(\alpha, f(\alpha))$ in $\mathbb{F}_{q^m}$ such that $\alpha$
is normal over $\mathbb{F}_q$ and
$\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\alpha^{-1})=a$, where $f(x)\in
\mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$. Further, when
$n=2$ and $q=5^k$ for some $k\in \mathbb{N}$, such a pair definitely exists for
all $(q, m)$ apart from at most $20$ choices.
