Recent studies show that a universal relation between black-hole (BH) growth and stellar mass ($M_bigstar$) or star formation rate (SFR) is an oversimplification of BH-galaxy co-evolution, and that morphological and structural properties of host galaxies must also be considered. Particularly, a possible connection between BH growth and host-galaxy compactness was identified among star-forming (SF) galaxies. Utilizing $approx 6300$ massive galaxies with $I_{rm 814W}~<~24$ at $z$ $<$ 1.2 in the COSMOS field, we perform systematic partial-correlation analyses to investigate how sample-averaged BH accretion rate ($rm overline{BHAR}$) depends on host-galaxy compactness among SF galaxies, when controlling for morphology and $M_bigstar$ (or SFR). The projected central surface-mass density within 1 kpc, $Sigma_{1}$, is utilized to represent host-galaxy compactness in our study. We find that the $rm overline{BHAR}$-$Sigma_{1}$ relation is stronger than either the $rm overline{BHAR}$-$M_bigstar$ or $rm overline{BHAR}$-SFR relation among SF galaxies, and this $rm overline{BHAR}$-$Sigma_{1}$ relation applies to both bulge-dominated galaxies and galaxies that are not dominated by bulges. This $rm overline{BHAR}$-$Sigma_{1}$ relation among SF galaxies suggests a link between BH growth and the central gas density of host galaxies on the kpc scale, which may further imply a common origin of the gas in the vicinity of the BH and in the central $sim$ kpc of the galaxy. This $rm overline{BHAR}$-$Sigma_{1}$ relation can also be interpreted as the relation between BH growth and the central velocity dispersion of host galaxies at a given gas content, indicating the role of the host-galaxy potential well in regulating accretion onto the BH.
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