District boundaries have long been sacrosanct. They prevent urban students, for example, from enrolling in suburban schools that typically have higher-income families and sometimes more lavish budgets.

Once thought to be moribund, the voucher movement was revived by gains Republicans made in the 2010 midterm elections. Fourteen states since then have introduced or expanded private school vouchers, according to the Friedman Foundation for Educational Choice.

The money for the vouchers would come from two federal programs that Mr. Romney would overhaul that target students deemed in need of extra support: Title 1, for economically disadvantaged students; and the Individuals with Disabilities Education Act. Currently the money from both programs, the largest K-12 initiatives in the Education Department, is awarded to states and districts based on federal formulas.

Grover J. Whitehurst, a Romney adviser, said that remaking the programs into individual payments that follow the student — he used the metaphor of a student’s backpack — could attract other streams of education dollars.

“If you connected state funding with federal funding, then you’re talking about a backpack with enough money in it to really empower choice,” said Mr. Whitehurst, director of education policy at the Brookings Institution. “The idea would be the federal Title 1 funds would allow states that want to move in this direction to do so, and if they did so, all of a sudden it’s a game changer.”