  In this paper we mainly investigate the Cauchy problem of a two-component
Novikov system. We first prove the local well-posedness of the system in Besov
spaces $B^{s-1}_{p,r}\times B^s_{p,r}$ with
$p,r\in[1,\infty],~s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ by using the
Littlewood-Paley theory and transport equations theory. Then, by virtue of
logarithmic interpolation inequalities and the Osgood lemma, we establish the
local well-posedness of the system in the critical Besov space
$B^{\frac{1}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1}$. Moreover, we present two
blow-up criteria for the system by making use of the conservation laws.
