Currently, the needles are only available at pharmacies with a prescription.
Under the bill, anyone could buy up to 10 at a time.
The bill's sponsors say that access to clean needles will help stem the spread to HIV and other blood-borne diseases. But opponents worry about the state legitimizing illegal drug use.
In 2008, New Jersey became the last state in the country to offer addicts some way to get clean needles legally. But the law allowed only limited needle-exchange programs in a few cities.
The Assembly's health committee is scheduled to consider the bill on Monday.