The bill unanimously passed its first hearing in a Senate committee Wednesday. The full Senate will consider it later.
Dozens of dog owners rallied at the Capitol for the bill. Some choked up when they testified in favour of the measure and recalled how their pets had been killed.
The proposal would require sheriff's and police departments to provide three hours of online training for deputies and officers on how to recognize dog behaviours and employ nonlethal methods to control them when necessary.
The County Sheriffs of Colorado worked on the legislation and supports it.