The Assembly is scheduled to take up the bill Monday afternoon.
It would make possession of 15 grams or less of marijuana a civil penalty rather than a criminal one.
Fines would range from $150 to $500, with $50 going toward drug education.
The measure has bipartisan support. But Gov. Chris Christie said at a town hall event last week he would veto the bill.
Christie says he's focused on implementing a strict state medical marijuana program. He says easing pot laws would be contrary to that goal. He also says the state shouldn't undercut federal drug laws.
The state Senate has not considered the measure.