The proposed amendment also includes a provision to allow a director to stay in the position for nine straight years.
The Bank Company (Amendment) Bill, 2017 was placed in parliament by Finance Minister Abul Maal Abdul Muhith on Tuesday.
The bill was then forwarded for scrutiny by the parliamentary committee on the law ministry with a report deadline of 60 days.
The existing law allows maximum two members of a family to stay on as directors of one bank. It allows an individual to work as bank director for two consecutive three-year terms. They must then take a three-year break before becoming director again.
Several stakeholders of the sector have criticised the move ever since the draft law received the cabinet's approval on May 8. The amendment's critics believe it would pave the way for nepotism in the private banking sector.
They have also accused it of trying to meet the needs of a few influential businesspersons. Several directors were at risk of being removed from their position because of the existing law.
Opposition Jatiya Party MP Fakhrul Imam opposed the bill on Tuesday, but the motion was eventually rejected in voice vote.