Lawmakers voted, 42-0, on Thursday to advance the proposal designed to limit frivolous claims. Omaha Sen. Scott Lautenbaugh said his bill still gives inmates the right to file.
He said 21 other states have enacted similar laws.
The bill was supported by the Nebraska Attorney General's office and county prosecutors, but opposed by the Nebraska Criminal Defense Attorneys Association and ACLU Nebraska.
Omaha Sen. Burke Harr, a former prosecutor, says the post-conviction relief process is often abused.
Supporters say meritless claims have clogged Nebraska's courts. The bill requires two more readings before it advances to the governor.