The state Legislature voted 41-6 Wednesday to give second-round approval to the limit. A final vote is expected Friday and then the bill will go to Gov. Dave Heineman, who has said he would support a 30-day age limit.
If Heineman signs the bill, it would become effective immediately.
"The rails are greased and the train's heading down the track," said state Sen. Tom Carlson.
Every state has a safe-haven law meant to protect newborns from being dumped in trash bins and worse, but Nebraska's is the only one that lacks an age limit. Thirteen other states have 30-day age limits.
Most of the children dropped off at Nebraska hospitals since the law took effect in July have been preteens or teenagers as old as 17.
A 15-year-old girl was left at a hospital by a relative, making her the 35th child abandoned under the law, state officials said Wednesday. The girl, a Nebraska resident, was dropped off Tuesday at a hospital in Grand Island. She was placed in a foster home. Further details were not immediately released.
Hospital officials have said some of the children cried hysterically for their parents or guardians not to leave.
Child experts and others worry that children old enough to be aware of what was happening to them could be scarred emotionally for life.
"They did not ask to come into this world, they did not choose the parents they may have been cursed with," Sen. Ernie Chambers said Wednesday.
However, many lawmakers, health officials and child welfare experts have been sympathetic to the parents and guardians, saying they were trying to get help for their troubled children and were not merely abandoning them.
Parents who have spoken publicly said they thought they had nowhere else to turn to get help for out-of-control, sometimes mentally ill children.
Lawmakers plan to address the lack and accessibility of services for troubled, older youths during the regular legislative session, which begins in January.