We studied the initial reaction mechanism of Zn precursors, namely, di-methylzinc ($Zn(CH_3)_2$, DMZ) and diethylzinc ($Zn(C_2H_5)_2$, DEZ), for zinc oxide thin-film growth on a Si (001) surface using density functional theory. We calculated the migration and reaction energy barriers for DMZ and DEZ on a fully hydroxylized Si (001) surface. The Zn atom of DMZ or DEZ was adsorbed on an O atom of a hydroxyl (-OH) due to the lone pair electrons of the O atom on the Si (001) surface. The adsorbed DMZ or DEZ migrated to all available surface sites, and rotated on the O atom with low energy barriers in the range of 0.00-0.13 eV. We considered the DMZ or DEZ reaction at all available surface sites. The rotated and migrated DMZs reacted with the nearest -OH to produce a uni-methylzinc ($-ZnCH_3$, UMZ) group and methane ($CH_4$) with energy barriers in the range of 0.53-0.78 eV. In the case of the DEZs, smaller energy barriers in the range of 0.21-0.35 eV were needed for its reaction to produce a uni-ethylzinc ($-ZnC_2H_5$, UEZ) group and ethane ($C_2H_6$). Therefore, DEZ is preferred to DMZ due to its lower energy barrier for the surface reaction.