Oklahoma’s conservative Republican governor, Kevin Stitt, and its conservative Republican schools superintendent have appeared to stand shoulder-to-shoulder in the years of Donald J. Trump, with the former sending Oklahoma guardsmen to the southern border and the latter stocking the state’s schools with Trump-branded Bibles.

But when the superintendent, Ryan Walters, proposed finding the undocumented students in Oklahoma’s schools, Mr. Stitt said enough is enough.

“When I saw them picking on kids, I thought that’s a step too far,” Mr. Stitt said in a recent interview in his office in the State Capitol.

In an era of anything-goes politics on the nation’s right, the fight in Oklahoma may suggest there is an outer limit to what is acceptable, even for conservatives. Or it could offer a preview of the next frontier in the nation’s battle over immigration.

“It’s incredibly unfortunate that the governor has decided to undermine President Trump’s immigration agenda and take these type of swipes at him,” Mr. Walters said. “He’s attacking President Trump.”

It began when Mr. Walters, the state’s elected superintendent of schools, proposed new rules that would require Oklahoma public schools to collect citizenship information from students. The proposed rules were approved by the state school board in January, and they are now being considered by the Oklahoma Legislature.