Under what circumstances may we ask to see a Social Security card?

Q: “We have had several employees over the years who completed our new hire forms (and front page of I-9) and transposed numbers in their Social Security numbers, which caused issues with Payroll. Can I require new hires to show me their Social Security cards so that I can confirm the number is correct? In most cases I will see this as one of the documents for the I-9, but if they choose other forms for their I-9, can I still ask? Can I make a copy if it for my payroll records? What if they refuse? Can I terminate them? What if they say they don't have a card or lost it?" – Michelle, Pennsylvania