The plan is to add an amendment to the Constitution through a convention - a longshot effort that has never been successfully done. All 27 amendments that have been adopted were proposed by Congress.
A balanced budget amendment is a core goal of conservative Republicans who have gained control of an increasing number of state Legislatures in recent years, now holding both chambers in 32 states. Backers include groups like the American Legislative Exchange Council and the Koch brothers-backed Americans for Prosperity.
The effort also comes against the backdrop of deep turmoil in Washington over debt spending. Top congressional Democrats last week cut a deal with President Donald Trump to increase the federal debt limit, avoiding for now a fight that commonly causes divisions and threats of a government shutdown.
The goal of amendment backers is to eliminate the federal deficit and drive down the national debt, which is approaching $20 trillion. The current federal budget includes spending of about $4 trillion and has a shortfall of nearly $700 billion, according to the Congressional Budget Office.
Congress debated a balanced budget amendment in the early and mid-1990s, but it did not pass.
The lawmakers meeting this week are discussing a process that requires several steps. Thirty-four states must vote to adopt the amendment and convene a convention, but it still must be ratified by three-quarters of the states. Now, 27 states have active requests to convene a convention, all controlled by Republicans.
Arizona state Rep. Kelly Townsend addresses delegates to a balanced budget planning convention at the Arizona Capitol in Phoenix, Sept. 12, 2017.
Arizona is hosting 75 delegates this week, all Republicans. Arizona state Rep. Kelly Townsend said efforts to invite Democratic states have not been successful. Proposed rules say delegates must be approved by both chambers of their state Legislature 'so that they can legitimately vote and represent their state,' Townsend said.
Opponents of the amendment argue that a convention could go dangerously off-track and move into wholesale rewrites of other areas of the Constitution, such as gun rights, an abortion ban and term limits. They also say a balanced budget amendment could threaten the economy.
'By requiring a balanced budget every year, no matter the state of the economy, such an amendment would risk tipping weak economies into recession and making recessions longer and deeper, causing very large job losses,' according to a policy paper by the liberal Center on Budget and Policy Priorities.
That's because lawmakers would be forced to cut spending during recessions, removing a key way the federal government can boost economic activity.
Townsend said the three-quarters requirement to ratify the amendment limits the chances of a 'runaway convention' where delegates could do a wholesale rewrite of the Constitution.
'Whatever we do when we close down and adjourn, our final product has to be viable. It's not binding yet, and the states have to ratify it - that's 38 of them,' she said.
Even some conservatives worry about a constitutional convention.
U.S. Rep. Andy Biggs, an Arizona Republican, routinely blocked legislation authorizing a convention during the four years he led the state Senate. He wrote a book in 2015, The Con of the Con Con, laying out his concerns about a convention.
Biggs wrote that if people believe the Constitution is fallible, 'how do you know that the remedy you rely on, Article V, is not flawed as well?'