Those detained for questioning include two party members, one from the opposition BNP party and a local leader of the Jamaat e Islami Islamist party.
The tailor, Nikhil Joarder, is the latest victim in a series of similar attacks in this South Asian nation.
Police said Joarder was inside his tailoring shop in Tangail on Saturday when at least two assailants drove up on motorbikes and attacked him with machetes. He died immediately, according to Tangail Police Superintendent Mohammed Tanvir.
ISIS has claimed responsibility for the attack, according to the nongovernmental counter-terror monitoring organization SITE.
The superintendent said a complaint was filed against Joarder in 2012 for allegedly making derogatory comments about the Prophet Mohammed. Joarder was then arrested and released after spending a few weeks in jail, police said. The three people detained were behind that case.
Despite the apparent claim of responsibility by ISIS, authorities said they didn't know "if there is a link between the murder and the comments he made in 2012," Tanvir told CNN. He said Monday that police were aware of the claim but that it was "premature" to comment on it.
"We're not brushing aside other possibilities like personal enmity," Tanvir added.
Spate of attacks
According to Marcia Bernicat, U.S. ambassador to Bangladesh, there have been at least 35 similar attacks carried out in the span of 14 months. She said of those, 23 have been claimed by a terrorist group.
The government of Bangladesh is accusing the country's two main opposition parties, the BNP and Jamaat-e-Islami, of being behind the assassinations.
"They have launched planned murders in the country," said prime minister Sheikh Hasina in a speech on Saturday, according to
the state news agency BSS.
Ruhul Kabir Rizvi, BNP's Joint Secretary General, rejected the accusations, calling them "fabricated" and "baseless."
Speaking to CNN on Monday, Rizvi accused Hasina of failing to arrest militants who are targeting civil society in Bangladesh.
"(The) government are not being able to handle the situation (and) catch the culprit. And that is why they are blaming BNP... to confuse the people and cover up their ineffectiveness."
LGBT activists killed in hacking attack Monday
Earlier this week, two LGBT activists -- one of whom also worked for the USAID, an American government organization for poverty prevention -- were hacked to death in Dhaka.
The USAID said the employee was Xulhaz Mannan, who worked at the organization but also served as editor of Bangladesh's first LGBT magazine.
The other victim was Tanay Mojumdar. Like Mannan, he was openly gay and was a leader in the fight for LGBT rights, a British photographer told CNN's Ivan Watson.
Ansar al-Islam, the Bangladeshi division of al Qaeda in the Indian subcontinent, has claimed responsibility for the killings.
Human Rights Watch called on Bangladesh to investigate the murders.
Bloggers, others targeted in hacking deaths
The killing of Joarder and Monday's attack follows several others in Bangladesh since 2013, including the hacking death of a professor at a bus stop April 23 and the killings of several bloggers -- six in the past 12 months alone.
The professor, 58-year-old Rezaul Karim Siddique, was an English teacher at Rajshahi University. A university student was detained in his death but not charged, according to Rajshahi police Commissioner Mohammad Shamsuddin.
It's not clear why Siddique had been stabbed in the neck as he awaited a bus to take him to campus.
"He was neither a blogger nor an anti-Islamic campaigner, but the pattern of the murder indicates Islamist militants involved in the recent spate of killings of secular bloggers might have a link," Shamsuddin said.
ISIS claimed responsibility for Siddique's death, saying he was slain "for calling to atheism." CNN could not independently confirm either the terror outfit's claim or Siddique's religious beliefs.
Extremists have 'hit list'
Reports of hacking deaths go back to 2013 in Bangladesh, most of them targeting bloggers. Since last year, Nazimuddin Samad, Faisal Arefin Dipan, Ananta Bijoy Das, Niloy Neel, Washiqur Rahman and Avijit Roy are among the writers who have been killed.
In September, Islamic extremists said they wanted to take their war against secular writers beyond Bangladesh's borders. They released a "hit list" of people they would target in Europe and North America.
"Let Bangladesh revoke the citizenship of these enemies of Islam," a statement accompanying the list says. "If not, we will hunt them down in whatever part of God's world we find them and kill them right there."
Amnesty alleges that Bangladesh authorities have done little to discourage such attacks. Exiled lesbian, gay, bisexual and transgender activists told the watchdog group that when they attempted to report threats to authorities, police responded by warning them they could be charged with "unnatural offenses."
Rather than offer security, Amnesty said, police have warned LGBT activists to be "less provocative."