The bill would change the Kentucky Constitution and would have to be approved by voters before it became law. It also requires a 60 percent vote in both the House and the Senate.
A similar effort failed to pass the legislature last year. The Democratic-controlled House favors an automatic restoration of voting rights while the Republican-controlled Senate favors a waiting period of three to five years.
U.S. Sen. Rand Paul, a likely Republican candidate for president, has urged state lawmakers to pass the bill.