"Why didn't the FBI, when it saw that a member of the American military was communicating with a radical Islamist cleric, open up an investigation of that member of the military? That's a big question that we want to answer," Lieberman said on Thursday.
Hasan has been charged by the military with 13 counts of premeditated murder and 32 counts of attempted murder at the Texas base where he was counselling troops.
The case drew heavy criticism after it was revealed that military investigators and FBI agents knew a year before the shooting that Hasan had been in contact with Anwar al-Awlaki, an anti-American Muslim figure sympathetic to al-Qaeda, but dismissed the contact as research.
The justice and defence departments officials sent a joint letter on Monday to Lieberman and Collins telling them that turning over the information and its disclosure could compromise the case against Hasan.
They added that making FBI and defence department agents available for interviews was against long-standing policy.
Collins and Lieberman rejected claims that the information could compromise the pending prosecution of Hasan.
"We owe it to [the victims] and their survivors and everyone else in our country to determine whether our government could have prevented their deaths - and if so, why it did not - so that we can make sure it does the next time," Lieberman said.
Lieberman and Collins said the defence department and FBI had turned over some documents to their committee but they primarily involved background material.