As of Tuesday, the legislation that contains measures specifically addressing victims who are Native American or Alaska Native has 58 supporters in the Senate, including its sponsor Sen. Patrick Leahy of Vermont.
Backing from 60 senators would mean a filibuster could be ended if one were launched against it.
Leaders of tribal nations are meeting with members of Congress this week as part of a legislative conference.
One of the provisions in the act would allow tribes to prosecute offenders who are not Native American or Alaska Native when their victims are and the violence happens on a reservation.
The lawmakers are seeking the support as Congress is embroiled in controversy over women's rights.