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For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le\|A\|w(B)$ holds, where $w(\cdot)$ and $\|\cdot\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\|A\|=\|A^k\|=1$ for some $k$, $1\le k