WASHINGTON — In a case that could reshape American politics, the Supreme Court appeared split Tuesday on whether Wisconsin Republicans gave themselves an unfair advantage when they drew political maps to last a decade.

If Justice Anthony Kennedy, whose vote almost certainly controls the outcome, is prepared to join his liberal colleagues, the court could rule for the first time that districting plans that entrench one party’s control of a legislature or congressional delegation can violate the constitutional rights of the other party’s voters. That could lead to changes in political maps across the country.

While both parties seek maximum partisan advantage when they can, Republicans controlled more state governments after the 2010 census and aggressively used redistricting to lock in electoral advantages to last for the next 10 years.

Kennedy suggested, as he did in another redistricting case 13 years ago, that courts perhaps could be involved in placing limits on extremely partisan electoral maps. But he did not tip his hand about whether the Wisconsin map that favors Republicans crossed a constitutional line.

Throughout the session, the justices and lawyers alike appeared to cast their questions and remarks with the hope of attracting Kennedy. He wrote in 2004 that he would be open to ruling for the challengers if the court could be shown a good way to measure and manage excessively partisan districts.

Paul Smith, the same lawyer who failed to get Kennedy’s vote and thus a majority 13 years ago, said technology and data analysis had so improved since then that there are good ways to measure when one party gives itself an unfair edge in creating districts.

Without the court’s intervention, Smith said on behalf of the Democratic voters, the next round of redistricting after the 2020 census will see far more extreme partisan maps.

“You are the only institution in the United States that can solve this problem just as democracy is about to get worse,” Smith said.

Chief Justice John Roberts worried about involving the Supreme Court in a glut of partisan redistricting claims that would follow if the Wisconsin Democrats prevail.

“We’ll have to decide in every case whether the Democrats win or the Republicans win,” Roberts said, a scenario that he said would damage the court’s credibility.

But Justice Ruth Bader Ginsburg said that a decision upholding the Republican-drawn districts in Wisconsin would encourage one party’s lawmakers to stack the deck against their opponents when they control the process and reduce the number of legitimately contested elections.

“What becomes of the precious right to vote?” she asked.