The men allegedly support another terror group, Al-Qaeda, and have links with Ansarullah, a terrorist group in Yemen, NIA sources said. They operate in different names including Wahadat-e-Islam, Jamaath Wahadat-e-Islam Al Jihadiye and Jihadist Islamic Unit.
The UAE had kept them in jail for six months before deporting them to India last week.
On Monday, the probe agency flew the men to Chennai, where a court sent them to custody till July 25.
Investigators say the group, many of them management professionals, had made the UAE their home. One of them had been staying in Dubai for 32 years.
"They have knowingly raised funds to use for terror attacks. The ideology is to establish ISIS in India by waging a war against the government of India," said NIA public prosecutor CS Pillai.
Following their interrogation, the NIA arrested two men in Nagapattinam - Harish Muhammad and Hassan Ali. Sources alleged Hassan Ali is an ISIS operative who has been recruiting people to strike India.
He also posted videos asking his supporters to launch strikes using vehicles as weapons, or with explosives, poison and knives, sources said.
The NIA has been cracking down on alleged ISIS groups for the last few months.
Five men were arrested in Coimbatore over alleged links with the suspected suicide bomber behind the Easter attacks in Sri Lanka that killed more than 200 people. Their families claimed it was a targeted action on Muslim groups.