Reaction energies were determined for reductive ring-opening reactions of Li$^+$-coordinated ethylene carbonate (EC) and vinylene carbonate (VC) by a density functional method. We have also explored the ring-opening of Li$^+$-EC and Li$^+$-VC by reaction with a nucleophile (CH$_3$O$^-$.) thermodynamically. Our thermodynamic calculations led us to conclude that the possible reaction products are CH$_3$OCH$_2$CH$_2$OCO$_2$Li (O$_2$-C$_3$ cleavage) for Li$^+$-EC +CH$_3$O$^-$., and CH$_3$OCHCHOCO$_2$Li (O$_2$-C$_3$ cleavage) and CH$_3$OCO$_2$CHCHOLi (C$_1$-O$_2$ cleavage) for Li$^+$-VC +CH$_3$O$^-$.. The opening of VC would occur at the C$_1$-O$_2$ side by a kinetic reason, although the opening at the O$_2$-C$_3$ side is more favorable thermodynamically.