SINGAPORE—A tariff agreement with Vietnam emphasizes one of the White House’s top priorities in this frantic round of dealmaking over global trade: Slamming shut any back door routes for Chinese goods to enter the U.

Vietnam finds itself caught between its two biggest trading partners as the Trump administration tries to stop Chinese exporters using Vietnam as a way station.

Vietnam was the retail industry’s backup plan when Trump first imposed duties on China, so some executives are relieved to see a potential deal with 20% levies.