No vote has been scheduled on the latest bill passed by the Senate.
"The Assembly continues talking to the key stakeholders involved and remains committed to resolving this matter in a timely fashion - as long as it's negotiated in good faith and meets the Assembly's commitment to fairness for everyone," Assembly Democratic spokesman Tom Hester said in a statement Tuesday.
The Assembly bill, sponsored by Speaker Vincent Prieto (D., Hudson), would allow arbitrators to award annual raises of up to 3 percent, if they included savings realized through increased employee contributions to health insurance premiums and reductions in force.
Christie eliminated those provisions and others in his veto.
In his veto message, Christie said exceptions to the cap "will necessarily impact the ability to provide other necessary municipal services," given that towns must also comply with a separate 2010 law that imposes a 2 percent limit on property-tax hikes.
Supporters of the governor's proposal point to findings by a task force that studied the law, which show that arbitration awards subject to the cap since it took effect had an average annual salary base increase of 1.9 percent. Contracts settled before reaching arbitration have also had smaller raises.
Before the implementation of the cap, raises frequently reached double digits, said William Dressel, executive director of the New Jersey State League of Municipalities.
In addition to driving down costs, supporters say, the cap has slowed property-tax growth. The tax levy grew 1.7 percent last year, according to the Department of Community Affairs. In the decade before his arrival, Christie says, property taxes increased an average of 7 percent annually.
"This is absolutely essential for mayors and governing bodies to try to keep a lid on property taxes," Dressel said.
Messages left with state firefighters' and police unions were not returned Tuesday afternoon.