Kaine said a "substantial majority" of families of victims of the Virginia Tech shootings agreed to the settlement.
Peter Grenier and Douglas Fierberg, who represent 21 families, also confirmed the settlement but would not discuss its terms until final papers are drawn in a few days.
Kaine spokesman Gordon Hickey described the settlement as "a work in progress."
"We're discussing things with the families. There's still a confidentiality agreement we're honoring until it's signed and in place," Hickey said.
Attorney General Bob McDonnell's office had no comment on the settlement, spokesman Tucker Martin said.
Seung-Hui Cho, a mentally disturbed student, killed 32 victims and wounded two dozen others at Virginia Tech on April 16, 2007 before committing suicide. Twenty-two families had previously filed notice with the state that they may sue.
It was not immediately clear whether the settlement was significantly different from an earlier state proposal, the details of which were obtained last month by The Associated Press. That proposed deal totaled roughly $8.5 million plus the cost of reimbursing and paying for medical and psychological treatment for victims' families and survivors.
That proposal called for representatives of each of those killed to receive $100,000. A pool of $800,000 was set aside for the injured in the plan, with individuals eligible to receive up to $100,000 apiece. Families of those killed could seek additional money from a $1.75 million hardship fund, and other money was to be set aside for attorneys' fees and a fund for charities.
The settlement also would give the injured and victims' families a chance to meet with the governor and university officials several times to discuss the mass shootings and changes on campus since then.
By accepting the proposal, family members gave up the right to sue the state government, the school, the local governments serving Virginia Tech and the community services board that provides mental-health services in the area.
The student gunman had been ruled a danger to himself during a court commitment hearing in 2005 and was ordered to receive outpatient mental health care, but never received treatment.
In October, the families and surviving victims received payments ranging from $11,500 to $208,000 from the Hokie Spirit Memorial Fund, set up in the days after the shootings to handle donations that poured into the school. That fund will remain open for contributions to scholarships for five years.
Families originally were told they had to respond to the state's offer by March 31, but the deadline was extended.