The restriction is designed to stop parents and guardians from using the law to drop off older children at Nebraska hospitals.
The state's safe-haven law is the only one in the country without an age limit and has led to the drop-off of 34 children -- none of them infants -- since July. Nebraska was the last state to adopt a safe-haven law.
Supporters say the age limit would reflect the original intent of the law -- to prevent newborns from being abandoned in trash bins or worse.
The drop-offs of older children under the law prompted many legislators to suggest services for troubled youths are inadequate.
"A human services system devoid of money and leadership ... over the course of many years and administrations brings us to this point," said Sen. Danielle Nantkes.
Senators vowed Tuesday to address the problem during the regular legislative session that begins in January. The age-limit measure advanced by lawmakers faces two more votes but doesn't appear to face any major obstacles.
Gov. Dave Heineman has said he would support an age limit anywhere from 3 to 30 days.
If it is eventually approved, Nebraska's safe-haven law will go from being the most unusual in the country to one of the most common. Thirteen states have a 30-day limit.
State Sen. Arnie Stuthman noted that Nebraska's open-ended law invited abandonments from out of state. Five of the 34 children dropped off since September have been from other states, he said.
On the Net:
Nebraska Legislature: http://www.nebraskalegislature.gov/