The Fairleigh Dickinson University PublicMind poll found 45 percent of registered voters in favor of the law and 40 percent opposed.
Assemblywoman Grace Spencer, a Democrat from Newark, introduced the bill in July. People letting their dogs or cats run free in their vehicles could be fined $20.
The poll found just over half of Democrats support it, while just over one-third of Republicans back the idea.
The telephone poll of 901 voters was conducted from Sept. 6-12 and has a margin of error of plus or minus 3 percentage points.