"I've been in regular and productive discussion with the governor's office and am confident we've reached common ground on this bill that, above all, protects taxpayers," he said.
Christie's spokesmen did not respond to a request for comment.
Assembly Minority Leader Jon Bramnick (R., Union) said: "This legislation is a key reform which helps local governments control costs and prevents property taxes from skyrocketing. Compromise is good government, and the speaker has shown leadership in resolving this issue."
Christie and others have said that the cap, signed into law in 2010, has helped slow property-tax growth because public safety workers' salaries take up a significant part of local budgets.
The Republican governor assailed Prieto and other Democrats for not renewing the law immediately, and local government officials warned that municipal budget costs would soon exceed revenues because a separate, permanent cap limits their ability to raise property taxes.
The Democrat-controlled Senate passed Christie's preferred version in March, but the Assembly, also controlled by Democrats, did not vote on that bill.
Instead, it passed a proposal that would allow for bigger raises in certain cases - for example, if towns realized savings through increased employee contributions to health benefits. The initial Assembly bill also would have exempted some unions from the cap.
Those provisions are no longer included. The new bill would, however, change the way the cap is calculated so that the awards are compounded annually for each year of the contract.
It also would give arbitrators more time to render decisions.