"On behalf of the loved ones who were violently taken from us, please reconsider your approach to large-capacity magazines as part of the comprehensive package of gun legislation," the families wrote in a letter to lawmakers.
Gov. Malloy backed the families.
"They've asked for an up or down vote on that provision, and, whether it's in the larger bill or as an amendment, the families, and every resident of our state, deserve a vote," he said in a statement.
"I have been clear for weeks that a ban on the possession and sale of high-capacity magazines is an important part of our effort to prevent gun violence," the governor added. "Simply banning their sale moving forward would not be an effective solution."
In addition, the bill would create the nation's first statewide registry for people convicted of crimes involving the use or threat of dangerous weapons. The registry would not be public, but available to law enforcement only. Furthermore, it would require eligibility certificates for the purchase of any rifle, shotgun or ammunition, and would significantly increase penalties for illegal possession and firearms trafficking.
Other parts of the bill establish safety standards for school building projects and require each school in the state to develop a safety and security plan. It also requires safe school climate committees to investigate instances of bullying and other threatening behavior. Other provisions address security at colleges and universities.
The bill allows school districts to require "mental health first aid" training for school personnel and creates a task force to examine the state's mental health system. Additionally, it alters state insurance regulations to beef up mental health care coverage.
Don Williams, the Senate president, told CNN he expected strong support for the bill from both sides of the aisle because of the bipartisan task force that put it together.
"Democrats and Republicans were able to come to an agreement on a strong, comprehensive bill," he added. "That is a message that should resound in 49 other states, and in Washington, D.C.. and the message is we can get it done here and they should get it done in their respective states and nationally in Congress."