Combine cream cheese, horseradish, and chives in a small bowl.
Pat each ham slice dry with a paper towel and spread a thin layer of the cream cheese mixture on top. Place 1 slice American cheese over each; you may need more than 1 slice to cover the ham. Spread another thin layer of cream cheese mixture on top; roll into a pinwheel.
Cover pinwheels with plastic wrap and chill in the refrigerator until firm, 1 hour to overnight. Trim off ends with a sharp knife and slice each roll into 1/2-inch slices.
