Around the country, the issue has proved to be a sure winner for the conservative base, with bills popping up in more than a dozen state legislatures to force future presidential candidates to prove their citizenship. Those legislatures, though, have been much more reluctant to turn this issue into concrete law.

Birther bills have foundered or fallen dormant in at least five states and are still being debated in more than a half-dozen others. In Arizona, where both legislative chambers passed one such bill, Gov. Jan Brewer, a Republican, vetoed it this week, calling it “a bridge too far.”

But now, Oklahoma, a deeply conservative state, could be the first to put its doubts into law, through a bill that would require all candidates, from town council hopeful on up, to prove that they meet the legal requirements for office. Those requirements vary by office, but presidential aspirants would need to, among things, file certified proof that they were born in the United States. The bill does not specify what documents would need to be filed. A vote was expected by next week.

Supporters of the measure, and others like it from Georgia to Montana, protest that they are not birthers, as doubters of Mr. Obama’s country of birth have been called, sometimes derisively. They say that they simply want to clarify the status of all candidates and that Mr. Obama’s case has only sharpened the issue and illuminated what they call a glaring hole in statutes governing eligibility to run for office.

“It’s not a birther bill, it’s a common-sense bill,” said a lead sponsor, State Senator Rick Brinkley, a Republican from suburbs of Tulsa. “If you’re going to file for office, you should be willing to substantiate that you meet the qualifications.”

In state capitols, the debates have been reframed in the dry language of good government — a simple effort to clear the air, supporters say, for confused voters. And because many of the bills failed this year, supporters are renewing their legislative battle plans for next year, in the heart of a presidential campaign.

Meanwhile, even beyond Mr. Trump, divergent views among Republican governors have heightened a new sense that the debate over the issue remains unresolved. After Ms. Brewer vetoed the Arizona bill, for example, Gov. Bobby Jindal of Louisiana, a Republican, said he would enthusiastically sign a similar bill, should it reach his desk.

“If one state passes, and the Obama administration basically ignores the requirement and does not qualify for the ballot in that state, that would send a very strong signal that we have a situation in the United States where someone who is not eligible is occupying the White House,” said Mark Hatfield, a Republican state representative in Georgia whose own ballot bill failed to get through. If Oklahoma does not go forward, and an override of Ms. Brewer’s veto in Arizona does not materialize, Mr. Hatfield said, “then other states, including Georgia, have a duty to step up.”

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