The Democratic governor said Tuesday that he's considering an executive order to make sure the information is in the computer system used by gun outlets, The Washington Post reported.
"I think there's a way to tighten this and to get more data onto the system," Kaine said in the Post report.
Gunman Cho Seung-hui would have been barred from purchasing the handguns he used to kill 32 people and himself under the executive order because a court had found him dangerously mentally ill.
The Post reported that even lawmakers reluctant to limit gun ownership supported the idea.
"The murderer down at Virginia Tech never should have been able to purchase a gun," Del. Mark Cole, R-Fredericksburg, said to the Post. "Someone who's declared a threat to themselves or others should not be able to purchase a firearm."