Following passage in the House on Wednesday, the bills go to the Senate. If they pass there, they go to President Donald Trump's desk for signature.
The regulation, released last August, established a road map for states to set up IRAs that participants would fund through payroll deductions.
These IRAs are similar to the accounts already available to investors: They have contribution limits of $5,500 each year, plus $1,000 if you're 50 and over.
The rule that governs 401(k) plans — the Employee Retirement Income Security Act of 1974 — has kept states from moving forward with payroll deduction IRAs. This was because it wasn't clear how these state-based plans would be subject to ERISA.
"When something is subject to ERISA, it becomes expensive to maintain," said Marcia Wagner, managing director of The Wagner Law Group in Boston.
The rule from last summer would keep ERISA from pre-empting state law. These jurisdictions already have their own regulations that would apply to the IRAs.
"Employers will face a confusing patchwork of rules, and many small businesses may forgo offering retirement plans altogether," Rooney said in a statement when he co-sponsored the resolution. "Congress must act to protect workers and small businesses from these misguided regulations."
Retiree advocacy groups expressed concern over what the House passage of the two provisions might mean for jurisdictions that are weighing these IRA programs.
"That's the more critical question: What will happen next?" said Cristina Martin Firvida, AARP, director of financial security.
"The states that have enacted these programs are committed to the programs that they've legislated," she said. "The states that are considering bills, that have voted on them but haven't had them go to signature — we are concerned these resolutions will have a chilling effect on that activity."
AARP estimates 55 million individuals don't have a way to save for retirement from their regular paycheck.
Eight states have enacted legislation to build retirement savings programs for workers: California, Connecticut, Illinois, Maryland, Massachusetts, New Jersey, Oregon and Washington.
The Massachusetts plan is currently available for workers at small nonprofits, but the Bay State has introduced legislation that will make a program available to private-sector workers who don't have a 401(k).
California's Secure Choice program is one of the newest efforts to be put in place. Gov. Jerry Brown signed a bill into law last September, establishing the program.
The vote comes on the heels of recent controversy surrounding the Labor Department's fiduciary regulation, a measure that requires financial advisors to provide you with advice that is in your best interest.
In a presidential memorandum, Trump called on the department to review the regulation. The agency is now exploring options to delay its originally scheduled April 10 applicability date.
(This story has been updated to add comments.)