Ahmedur Rashid Tutul, 43, Ranadipam Basu, 50, and Tareque Rahim, 30, were taken to the hospital with severe injuries after they were hacked by unknown assailants at the office of Shuddhaswar publications in the capital.
"We sent them to the DMCH (Dhaka Medical College Hospital) for treatment," an officer of the nearby Lalmatia police station told PTI.
A resident of the building told reporters that the assailants also fired gunshots though police said victims' wounds were caused by knife attacks.
"The condition Tutul is critical," doctors at the hospital said.
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Tutul, who runs Shuddhaswar publications, had lodged a complaint with police after receiving death threats on Facebook following Roy's murder in February this year.
Four secularist bloggers have been killed by suspected Islamist militants in Bangladesh this year.
Bangladesh-born US writer Roy was hacked to death by unidentified assailants on February 26. Blogger Washiqur Rahman was murdered in central Dhaka on March 30. Writer and blogger Ananta Bijoy Das was killed in a similar attack in northeastern city of Sylhet on May 12.
Ansar Al Islam, a radical group linked to al-Qaeda, had claimed responsibility for the killings on social media, calling the victims enemies of Islam and Allah.