Evidence for deliquescence of perchlorate salts has been discovered in the Martian polar region while possible brine flows have been observed in the equatorial region. This appears to contradict the idea that bulk deliquescence is too slow to occur during the short periods of the Martian diurnal cycle during which conditions are favorable for it. We conduct laboratory experiments to study the formation of liquid brines at Mars environmental conditions. We find that when water vapor is the only source of water, bulk deliquescence of perchlorates is not rapid enough to occur during the short periods of the day during which the temperature is above the salts' eutectic value, and the humidity is above the salts' deliquescence value. However, when the salts are in contact with water ice, liquid brine forms in minutes, indicating that aqueous solutions could form temporarily where salts and ice coexist on the Martian surface and in the shallow subsurface.