It has been known that giant DNA shows structural transitions in aqueous solutions under the existence of counterions and other polymers. However, the mechanism of these transitions has not been fully understood. In this study, we directly observed structures of probed (dye-labeled), dilute DNA chains in unprobed DNA/polyethylene glycol (PEG)/monovalent salt (NaCl) aqueous solutions with fluorescent microscopy to examine this mechanism. Specifically, we varied the PEG molecular weight and salt concentration to investigate the effect of competition between the depletion and electrostatic interactions on the coil-globule transition and the aggregate formation. It was found that the globules coexist with the aggregates when the unprobed DNA chains have a concentration higher than their overlap concentration. We discuss the stability of the observed structures on the basis of a free energy model incorporating the attractive depletion energy, the repulsive electrostatic energy, and the chain bending energy. This model suggested that both of the globules and aggregates are more stable than the random coil at high salt concentrations/under existence of PEG and the transition occurs when the depletion interaction overwhelms the electrostatic interaction. However, the coexistence of the globule and aggregate was not deduced from the thermodynamic model, suggesting a nonequilibrium aspect of the DNA solution and metastabilities of these structures. Thus, the population ratio of globules and aggregates was also analyzed on the basis of a kinetic model. The analysis suggested that the depletion interaction dominates this ratio, rationalizing the coexistence of globules and aggregates.