HONG KONG — The company that manages the Donald J. Trump brand has moved to protect the name in Macau, a part of China that long ago surpassed Las Vegas to become the world’s largest gambling market.

The moves do not necessarily indicate that President Trump or the Trump Organization will eventually open a Trump hotel or casino there. But the ongoing efforts to expand Trump trademarks overseas have raised questions about whether they fall afoul of a constitutional prohibition on receiving economic benefits from foreign governments.

Since winning the presidential election, Mr. Trump has turned over control of his businesses to his adult sons, and his legal team has said the business would no longer pursue new foreign deals.

DTTM Operations, a New York company that holds Trump trademarks, applied in June for rights to the Trump name in casinos, construction, hotels and real estate, according to Macau government filings. It is not clear whether the applications were for new trademark protections or to renew similar trademarks dating back to 2006 that had expired.