  Given $F= \mathbb{F}_{p^{t}}$, a field with $p^t$ elements, where $p $ is a
prime power, $t\geq 7$, $n$ are positive integers and $f=f_1/f_2$ is a rational
function, where $f_1, f_2$ are relatively prime, irreducible polynomials with
$deg(f_1) + deg(f_2) = n $ in $F[x]$. We construct a sufficient condition on
$(p,t)$ which guarantees primitive pairing $(\epsilon, f(\epsilon))$ exists in
$F$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon) = a$ and
$Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon)) = b$ for any prescribed $a,b
\in \mathbb{F}_{p}$. Further, we demonstrate for any positive integer $n$, such
a pair definitely exists for large $t$. The scenario when $n = 2$ is handled
separately and we verified that such a pair exists for all $(p,t)$ except from
possible 71 values of $p$. A result for the case $n=3$ is given as well.
