Two major health-care deals that stand to reshape the insurance and pharmacy industries are moving toward winning antitrust approval, according to a person familiar with the matter, a final step as the takeovers head toward completion.

The U.S. Department of Justice is on track to soon approve the acquisition of pharmacy-benefit manager Express Scripts Holding Co. by health insurer Cigna Corp., as well as CVS Health Corp.’s takeover of insurer Aetna Inc., said the person, who asked not to be...