He posted a photo of the opened suitcase on Facebook on Monday and said the incident occurred when he was returning to the US on Sunday.
The suitcase, he said, contained some books on his father Tajuddin Ahmad, the first prime minister of Bangladesh.
"My name is clearly marked on the nametag," he said.
Dhaka Customs House Assistant Commissioner Md Saidul Islam passed the responsibilities on to the airlines.
"The law enforcers search someone's luggage for suspicious things in his or her presence. When the passenger hands the baggage to the airlines, it gives him or her a tag," he said.
Qatar Airways confirmed that Sohel Taj flew to the US with them.
Its customer service officer in Dhaka told bdnews24.com: "We've come to know about the matter. Many are calling us to inquire about the issue."
"He is a respected person. We will take necessary steps after investigation," the officer added.