A theoretical description of the time-resolved infrared-visible sum-frequency-generation (IV-SFG) process when the incident infrared pulsed beam is either left- or right-circularly-polarized (CP) is presented. Even for isotropic chiral molecular liquids, the CP IV-SFG polarization does not vanish because the chiral component of the associated response function is nonzero due to the breakdown of the Born-Oppenheimer approximation as well as to the finite polarizability-electric-quadrupole response function, which is a fourth-rank tensor. For a specific perpendicular detection scheme. it is shown that the three different contributions, originated from (1) all-electric-dipole-allowed polarization, (2) polarizability-quadrupole-induced polarization, and (3) interference between the above two polarizations, to the CP IV-SFG signal can be sperately measured. Also, the circular intensity difference, which is the difference between the left-CP IV-SFG intensity and the right-CP IV-SFG intensity, is theoretically investigated and found to be solely determined by the interference between the all-electric-dipole-allowed polarization and the polarizability-quadrupole-induced polarization. Thus, it is shown that the time-resolved CP IV-SFG methods are useful for the investigation of the molecular chirality of liquids and vibrational optical activity.