The poachers were carrying the cubs to West Bengal in two cartons in a jeep, Anisur Rahman, district police chief said.
Two people were arrested in this connection and the initial police investigations revealed that the wild animals were brought from Dhaka and the jeep was supposed to reach a house in a bordering village for the planned smuggling.
Police said they were investigating whether the tiger cubs were poached from the Sundarbans, the world's largest mangrove forest, which has a significant population of the endangered Bengal tigers.
Forest officials said they had planned to release them at Bangabandhu Safari Park on the outskirts of the capital.
Bangladesh brought in new laws in 2010 to protect wildlife, but poaching remains rampant.
A 2015 census found the number of Bengal tigers in the 10,000 sq-km Sunderbans to be just over 100 while the figure was estimated at 440 a decade ago.