Speaking on the House floor in 1993, then-Rep. Newt Gingrich termed the vote on the pending North American Free Trade Agreement a “magic moment” — a chance to strengthen ties with Mexico and to spur an economic boom in the U.S.

Now Mr. Gingrich is reportedly one of the short-listed candidates for ticketmate with the GOP’s presumptive presidential nominee, Donald Trump, who has staked his campaign on opposition to trade in general and NAFTA in particular, likening it to rape and calling it the “worst trade deal in the history of the country.”

Mr. Gingrich also supports granting legal status to illegal immigrants, has mocked Mr. Trump’s call for a ban on Muslims being admitted to the U.S. and backed the 2003 invasion of Iraq — all at odds with Mr. Trump’s major stances.

That he’s still one of the few names who keep popping up, along with New Jersey Gov. Chris Christie, underscored just how tricky the veep search is for Mr. Trump’s team, which must try to match their candidate’s brash style and populist politics with a GOP still very much in the grips of Bush-era Republicanism.

Craig Shirley, author of “Citizen Newt,” a biography of the former speaker, said the differences between Mr. Gingrich and Mr. Trump on trade reflects fundamental differences in their governing approaches.

“Newt is as much a Reagan conservative today as anybody in the national party,” Mr. Shirley said, pointing out that former President Reagan proposed a North American free trade accord in 1979. “Trump is not a Reagan conservative, Trump is a Nixon conservative.”

Less than two years after that 1993 floor speech, Mr. Gingrich would become the first Republican House speaker in four decades, would oversee welfare reform and laid the groundwork for a series of balanced budgets to close out the Clinton administration.

He himself was gone from office by then, damaged by the fallout from investigations into President Clinton’s misbehavior and impeachment, and ousted in the end by a conservative rebellion within his own party. He ran for president in 2012, winning a stunning victory in South Carolina’s primary but fading soon thereafter.

Mr. Gingrich, who writes a column for The Washington Times, did not return a phone call seeking comment.

But on Sunday fuel was added to Gingrich speculation when the former speaker won a Republican vice presidential straw poll of attendees at a major conservative gathering of more than 4,000 attendees that Mr. Trump himself had addressed Friday.

Mr. Gingrich took 194 of the 985 votes (20 percent) cast at the seventh annual Western Conservative Summit in Denver, followed by Sen. Tom Cotton, Arkansas Republican, who took 148 votes, or 15 percent.

Mr. Christie, meanwhile, received only 15 votes and failed to crack the top five.

The New Jersey governor is serving his second term, having flamed out in this year’s primaries and then quickly backing Mr. Trump.

During the primary campaign, Mr. Christie blasted Mr. Trump’s proposed ban on Muslims, mocked his plans to build a border wall and questioned his experience and temperament. Mr. Trump was just as critical of Mr. Christie, blasting his management of New Jersey.

Now, with two weeks to go until the convention, Mr. Trump is narrowing his potential picks and those bad feelings — and the discordant policy stances — are forgotten.

Over the weekend Mr. Trump also met with Indiana Gov. Mike Pence, who some conservative activists are pushing as a vice presidential pick.

It’s noteworthy that all three men have supported legalization of illegal immigrants — though Mr. Pence’s stance is perhaps closest to Mr. Trump’s since he calls for immigrants to go home, then apply to return, though perhaps in as fast as a week.

Conservative activists say they’re closely watching Mr. Trump’s pick, hoping the choice will ease their fears about the brash businessman.

But Terry Holt, a GOP consultant who worked on Capitol Hill and for the 2000 and 2004 Bush campaigns, said Mr. Trump’s choice has little to do with his chances for success.

“The vote in November will be about Trump,” he said. “Mickey Mouse could be the veep and it wouldn’t matter.”

Still, Mr. Shirley said someone like Mr. Gingrich has the potential to help Mr. Trump.

“He is a unifying force, and that is what Trump needs right now. He needs to unify the party,” Mr. Shirley said. “Plus, he knows Washington, but he is not part of the Washington culture. He is an original Reaganite, and having someone with binding ties to Reaganism will help Trump.”

Mr. Trump could announce his pick before the convention in hopes of ramping up excitement for his candidacy before thousands of Republicans converge on Cleveland.

Mr. Shirley said the pick will help set the tone for the rest of the race.

“It is astonishingly important because it unifies the convention or it divides the convention,” he said. “If it unifies the convention, they generally go on to win in the fall. If it divides the convention, they usually go on to lose.”

• Valerie Richardson contributed to this report from Denver.

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