Republicans control both chambers of the Legislature in the two states and will continue to do so next year.

“The root of this is that the clock is ticking, and you get it done now or it's going to be hard in some of those states with a Democratic governor,” said Nick Everhart, an Ohio-based Republican strategist, noting that legislators may not be so eager to consolidate power when they or their successors are representing more competitive districts in the next decade. “Who wants to take those votes in a potentially different world in ’21 or ’22?”

The more fundamental risk for Republicans is that voters are increasingly savvy about issues like gerrymandering, voting rights and campaign finance, which were once thought to be of great interest to elites but of little relevance to most citizens. Those are some of the issues at stake in the G.O.P. bills under consideration this month.

Many voters have become more supportive of a nonpartisan approach to redistricting, following years of gerrymanders that carved up states into impregnable red and blue districts and led to more political polarization rather than cooperation. Four states last month approved ballot measures that would create some form of an independent redistricting commission.

Some leading Republicans believe their party is tempting an even greater backlash than they saw in 2018 if they defy the wishes of an electorate increasingly attuned to fairness and angered by what figures as different as President Trump and Senator Bernie Sanders have denounced as “a rigged system.”

Frank LaRose, a Republican and Ohio’s newly elected secretary of state, called it “tough love for the party, but it’s what the voters are starting to expect and demand from us.” A former state senator, Mr. LaRose recalled that when he first began advocating for redistricting reform seven years ago, “folks’ eyes would roll back in their head, but it’s becoming more of a mainstream issue and it’s not just the League of Women Voters crowd or political science students.”