DNA G-quadruplex structures are emerging cancer-specific targets for chemotherapeutics. Ligands that bind to and stabilize DNA G-quadruplexes have the potential to be anti-cancer drugs. Lack of binding selectivity to DNA G-quadruplex over DNA duplex remains a major challenge when attempting to develop G-quadruplex ligands into successful anti-cancer drugs. Thorough understanding of the binding nature of existing non-selective ligands that bind to both DNA quadruplex and DNA duplex will help to address this challenge. Daunomycin and doxorubicin, two commonly used anticancer drugs, are examples of non-selective DNA ligands. In this study, we extended our early all-atom binding simulation studies between doxorubicin and a DNA duplex (d(CGATCG)) to probe the binding between daunomycin and a parallel DNA quadruplex (d(TGGGGT)) and DNA duplex. In addition to the end stacking mode, which mimics the mode in the crystal structure, a pure groove binding mode was observed in our free binding simulations. The dynamic and energetic properties of these two binding modes are thoroughly examined, and a detailed comparison is made between DNA quadruplex binding modes and DNA duplex binding modes. Implications on the design of more selective DNA quadruplex ligands are also discussed. Graphical abstract Top stacking and groov binding modes from the MD simulations.