Rescue efforts are under way as officials assess the damage and search for the missing people, Nepali authorities say.
An army helicopter was able to lift people stranded by the flooding.
Police said 95 rescuers were already at the area and more were expected to join in rescue efforts.
The flooding on the Bhotekoshi River destroyed the Friendship Bridge at Rasuwagadi, which is 120km north of the capital, Kathmandu, in the early hours of Tuesday.
The flooding also swept away several houses and trucks that were parked at the border for customs inspections.
Hundreds of electric vehicles imported from China were also parked at the border point.
The 18 missing are 12 Nepali citizens and six Chinese nationals, according to the National Disaster Risk Reduction and Management Authority.
The destruction of the bridge has halted all trade from China to Nepal through this route.
The longer alternative is for goods to be shipped from China to India and then brought overland to Nepal.
Monsoon rains that begin in June and end in September often cause severe flooding in Nepal, disrupting infrastructure and endangering lives.
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