Two private prison companies are facing legal pressure over work programs at detention centers that pay migrants $1 a day to perform tasks such as cleaning or working in the centers’ kitchens.

Recent lawsuits have accused Geo Group Inc. and Co Inc. of violating federal laws prohibiting forced labor, state minimum-wage laws or unjust-enrichment laws. The companies run detention centers for migrants caught crossing the southern border, detained asking for asylum at an official entry place or accused of living in the U.S. illegally....