The demolition of the sculpture has triggered a strong reaction across Kushtia.
Kushtia Superintendent of Police SM Tanvir Arafat said CCTV footage of last night showed two people vandalizing the sculpture at 2am.
"We are trying to identify them and they will be brought to justice soon."
To mark Bangabandhu's birth centenary, the local administration decided to set up three sculptures of Bangabandhu at the intersection of five streets in the city in November this year.
Also, it planned to build murals of the four national leaders at the bottom of the sculpture, Kushtia Municipality sources said.
The construction of one of the sculptures, Bangabandhu giving his March 7 speech, started on November 17 and was nearing completion, Kushtia Municipality Engineer Rabiul Islam said.
Meanwhile, the making of a sculpture of the Father of the Nation in the capital has recently stirred up a lot of controversies.
On November 13, Islami Andolan Bangladesh Nayeb-e-Ameer Faizul Karim and Hefazat-e-Islam Bangladesh Joint Secretary General Mamunul Haque opposed the setting up of the sculpture.
However, the law enforcement came down on anti-sculpture protesters in the city's Paltan intersection on Friday.
They brought out a rally from Baitul Mukarram National Mosque after Jumma prayers without Dhaka Metropolitan Police's (DMP) approval, the law enforcers said.
Earlier, ruling Awami League General Secretary Obaidul Quader made it clear that any attempt to destroy religious tolerance in the country would be dealt with strictly.
Also read: Police foil anti-sculpture protest outside Baitul Mukarram Mosque