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 identified because the process is private.

CVS and Aetna are in talks with the Justice Department about divesting Medicare prescription...