Senators gave the measure first-round approval Wednesday with a 39-2 vote.
The measure by state Sen. John McCollister, of Omaha, would require employers to give job seekers the opportunity to explain themselves if applicants are forced to disclose their criminal records. It would only apply to businesses with 15 or more employees.
McCollister says the bill is designed to help former convicts who often struggle to find work after their sentences end, while allowing businesses to avoid hiring inappropriate people for a job.