Aminul Islam Baig, 40, and Shakib bin Kamal, 30, were both arrested on Sunday night during raids at two addresses in Dhaka in which officers also recovered militant literature and computer equipment, police added.
Dhaka police spokesperson Jahangir Alam said both men were members of the banned militant outfit Jamayetul Mujahideen Bangladesh (JMB).
"They are active members of JMB and admitted their connection with IS," Alam told AFP.
During questioning, Baig and Kamal provided a list of 20 accomplices who had been helping them try and persuade young Muslims to join the thousands of foreigners already fighting alongside IS, which has declared a "caliphate" across vast parts of Syria and Iraq.
"They have been trying to recruit people all over the country to fight with the IS militants in Syria and other countries of the Middle East," Alam added.
Another police spokesperson, Monirul Islam, told a news conference in Dhaka that Baig was the head of IT at a multinational company in Bangladesh, without naming the organisation, while Kamal was a teacher at a renowned school.
Islam said the pair would be produced before court soon, adding that police would "seek remand for further interrogation".
Bangladesh is an officially secular country but more than 90 percent of its 160 million people are Muslim.
Although radical groups have struggled to make headway in the political mainstream, a recent spate of killings of secular bloggers has highlighted fears of growing radicalisation among some Bangladeshis.
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