Taken together, the two waiting periods would provide the nation’s undocumented workers with a path to United States citizenship in 13 years, matching the draft of a plan by President Obama to offer full participation in American democracy to millions who are living in fear of deportation.

The arrangement would shrink the amount of time it takes to become a naturalized citizen, from five years to three years. But in an appeal to Republicans, it would also extend to 10 years, from 8, the amount of time that illegal immigrants must wait before receiving permission to work in the United States permanently.

Such a compromise might give both sides something to crow about: Republicans could argue that they pushed for a longer waiting period before an immigrant could get a green card, which allows its holder to remain and work in the United States indefinitely. Democrats could say that undocumented workers would become citizens faster.

“It is an unusual construction, but it gets them to citizenship in the same time as the administration plan,” said Kevin Appleby, the director of migration policy at the United States Conference of Catholic Bishops. “Most importantly, it eliminates the prospect of a permanent underclass by ensuring that, in time, all will have the opportunity to become Americans.”