  In this paper, we show that the solution map of the two-component Novikov
system is not uniformly continuous on the initial data in Besov spaces $B_{p,
r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p},
\frac{3}{2}\}$, $1\leq p< \infty$, $1\leq r<\infty$. Our result covers and
extends the previous non-uniform continuity in Sobolev spaces
$H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{5}{2}$ (J. Math.
Phys., 2017) to Besov spaces.
