Hawzah News Agency (New York, USA) - The lawsuit, filed in a New York court, seeks compensation and punitive damages for the deaths of Alexander Pinczowski, 29, and Nohemi Gonzalez, 26.
Twitter has not commented on the case.
In March 2016, 32 people were killed in the Brussels attacks. The November 2015 attacks in Paris left 130 people dead.
ISIS militants have claimed responsibility for the assaults.
Mr Pinczowski, who was in Brussels on a business trip last March, was killed when a blast ripped through the check-in area of the airport in the Belgian capital.
The families accuse social media company Twitter of having "knowingly provided material support and resources to ISIS in the form of Twitter's online social network platform and communication services".
The plaintiffs allege that IS militants have "used and relied on Twitter's online social network platform and communications services as among its most important tools to facilitate and carry out its terrorist activity", including the attacks in Brussels and Paris.
The lawsuit says that Twitter has continued to provide such resources "despite receiving numerous complaints and widespread media and other attention for providing its online social media platform and communications services to ISIS ".
"Among social media platforms, Twitter has most brazenly refused to cut off its services to terrorists, taking the position that 'the tweets must flow' even if it means assisting in mass murders," the attorney said in a statement.
Twitter has so far made no public comments on the issue.
In February 2016, the US-based company said it had suspended more than 125,000 accounts since mid-2015 "for threatening or promoting terrorist acts".