U.S Customs regulations reads like this:
“There is no duty on gold coins, medals or bullion but these items must be declared to a Customs and Border Protection (CBP) Officer. Please note a FinCEN 105 form must be completed at the time of entry for monetary instruments over $10,000. This includes currency, ie. gold coins, valued over $10,000. The FINCEN definition of currency: The coin and paper money of the United States or any other country that is (1) designated as legal tender and that (2) circulates and (3) is customarily accepted as a medium of exchange in the country of issuance.”
It does not say whether the $10,000 value will be assessed by the face value of the coin, in the case of the American Gold Eagle this would mean that every ounce of gold has the value of $50 USD, or the market value of the gold, being around $1565.00 currently.