Let $A$ be the set of people who read newspaper A and $B$ be the set of people who read newspaper B. According to the problem, we have:
$|A| = 63$
$|B| = 76$
$|A \cup B| = |A| + |B| - |A \cap B|$
Also, $|A \cup B|$ represents the percentage of people who read at least one of the newspapers, and it cannot exceed $100$. We are given that $x$% of the people read both newspapers, so $|A \cap B| = x$.
Therefore, $|A \cup B| = 63 + 76 - x = 139 - x$.
Since $|A \cup B| \le 100$, we have $139 - x \le 100$, which implies $x \ge 39$.
From the Venn diagram, $x$ must be less than or equal to both $63$ and $76$. So, $x \le 63$.
Thus, $39 \le x \le 63$.
From the given options, the only value that satisfies this inequality is $55$.
Therefore, the possible value of $x$ is $55$.