The exchange occurred with about five minutes remaining in the first half of the Timberwolves' 129-114 victory at Target Center in Minneapolis.
Towns caught a pass in the paint at the start of the sequence, while defended by Gay and LaMarcus Aldridge. Towns backed Gay down toward the rim and used his left elbow to pound into his chest. Gay took exception to the physical contact and looked at a nearby referee for a potential foul call. Towns slammed the ball on the ground as the official called a foul on Gay.
Gay stood chest-to-chest with the Timberwolves center and the two stars shouted at each other before they were separated by teammates.
Towns scored 28 points and had 11 rebounds and four assists in the win. Andrew Wiggins led the Timberwolves with 30 points and DeMar DeRozan led the Spurs with 27 points. Gay had 10 points and eight rebounds for the Spurs.