The stampede erupted when hundreds of people tried to force their way into the factory compound through a small gate after massing outside the building before dawn, local police chiefs told AFP.
"We have so far recovered 23 bodies. Most of the dead are poor and emaciated women," Mymensingh police chief Moinul Haque told AFP.
Read: Ten killed in stampede during Hindu ritual in Bangladesh
Kamrul Islam, the senior officer at a police station near the factory, said the death toll could rise.
"Some people had taken the bodies of their relatives before police arrived at the scene," he said.
The owner of the factory and six others have been arrested, Islam added.
Police said up to 1,500 people had massed outside the chewing tobacco factory at around 4:45 am (2245 GMT on Thursday) after the plant owners had announced they would distribute free clothes to poor people in accordance with Islamic ritual.
Rich Bangladeshis often distribute free clothes to poor people during the Muslim holy month of Ramazan.
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