Vice President Joseph R. Biden, in a moment of levity, looked into the TV cameras at an Aspen, Colorado, event at which he and possible Donald Trump running mate Newt Gingrich were featured speakers on Saturday. As if addressing the presumptive Republican presidential nominee, Mr. Biden said of Mr. Gingrich, “Donald, I’m not endorsing him. He’s bright as hell, but I disagree with him.”

A wide assortment of Republican officials and activists not only share Mr. Biden’s esteem for Mr. Gingrich’s IQ but do what Mr. Biden didn’t do: give the former House speaker a nod as a vice presidential candidate, a slot Mr. Trump may fill this week.

Why Mr. Gingrich?

A reliable, knowledgeable and relatively colorless conservative like Indiana Gov. Mike Pence — said to be a favorite of Mr. Trump’s family — or an equally reliable conservative like Oklahoma Gov. Mary Fallin, also a former U.S. House member, would seem a safer, more logical choice than Mr. Gingrich.

Yet Mr. Trump and Mr. Gingrich, unlike in looks, vocabulary and personality, are a match in other ways.

The former House speaker was once the most disliked politician in America. Now Mr. Trump shares that title with Hillary Clinton: both having an ostensibly crushing 57 percent disapproval rating in the latest Quinnipiac University Poll.

More counterintuitive is Mr. Gingrich’s wide array of backers for the VP job, given that he came out on the wrong end of an ethics battle while in Congress, and then had to fight off at least one coup attempt by disgruntled fellow House Republicans before abruptly quitting as speaker and leaving Congress after the 1998 elections. Some Republicans had blamed his election strategy for the loss of five House seats and the failure to add any Senate seats for the party.

It didn’t seem to help his image at the time that, four years earlier, it was his election strategy, including his innovative “Contract With America,” that won credit with Republicans regaining control of the House after 40 years in the minority.

But again, why Mr. Gingrich now?

Idaho Republican Party Chairman Stephen Yates points to Mr. Gingrich’s “knowledge of budget, congressional processes in general and deep interest in national security.”

These factors “are a great complement to the strengths that won Donald Trump the nomination,” said Mr. Yates, a former National Security Agency analyst who later served as Vice President Dick Cheney’s deputy national security adviser.

Deputy House Majority Whip Tom Cole, Oklahoma Republican, says that while virtually nobody but family and friends care much who the No. 2 person is on the ticket, a “running mate can bring credibility, energy and a new weapon to attack the opposition. After that, a good pick may even help to govern if his party’s presidential candidate actually wins.”

Mr. Cole, a seven-term House member, suggests that by “those standards, Gingrich has a lot to offer — incisive, sharp communicator, boundless energy, will chop up Hillary every day and defend and explain Trump better than Trump can defend and explain himself.”

Should a Trump-Gingrich ticket win, they’d be a good match as a governing team, Mr. Cole said.

“No one knows D.C. better than Newt. And if Trump wins, he would be the ideal intermediary between Trump and the Hill, explaining each to the other,” Mr. Cole said. “Yes, he has baggage, but less baggage than Trump himself.”

Pastors and Pews founder David Lane, who trains preachers across the country to be Christian soldiers in conservatism’s cause, sees a certain uniqueness in Mr. Gingrich.

“Newt may be the only living former legislator who can walk in on Friday, Jan. 20, 2017, with the working knowledge to orchestrate and drive an agenda for limited government, deregulation of business, lower taxes and return of control to the states,” Mr. Lane said.

“Besides helping pull the wagon to get Trump elected, Newt may be the only adult in the room when it comes to governing with the institutional knowledge and grit to make the hard decisions to save America,” the Los Angeles-based Mr. Lane added.

Mr. Gingrich wants the running-mate job. Just as important, over the past several months of advocacy for Mr. Trump, Mr. Gingrich has shown that, despite what some regard as an intimidatingly large vocabulary, 40 years of experience in politics, quick thinking and sharp humor, he can manage to extol the billionaire businessman’s virtues and explain his failures without upstaging him.

Not only do Mr. Lane and Mr. Cole seem to sing from the same hymnal on why Mr. Gingrich qualifies for consideration, both the former House speaker and the presumptive nominee themselves seem to be hitting those same notes.

“Trump has said over and over again he needs a vice president who understands Washington, because he doesn’t,” the former House speaker and ex-history professor said at an Aspen conference on Saturday.

Pennsylvania GOP campaign adviser Charlie Gerow, who was partial to Sen. Ted Cruz of Texas for the presidential nomination, defined the downside and upside of Mr. Gingrich on the ticket.

“Newt offers the ability to get things done in Washington together with a world of policy initiatives, each of which would score big points for Trump,” Mr. Gerow said. “He’s also media-savvy and connects with voters as well as anyone. The only possible downside: At 73, he’s three years older than Trump, though age certainly hasn’t been on the minds of voters this year.”

Mr. Trump’s greatest need in a VP choice is someone who can help bring him a swing state like Pennsylvania, Ohio, Florida, Wisconsin, Iowa or Virginia, or turn a blue state like New Jersey red.

Pennsylvania has no plausible candidate; Ohio Gov. John Kasich has been too hard on Mr. Trump to make a matchup plausible; both former Gov. Jeb Bush and Sen. Marco Rubio have said they will not be a Trump running mate; and New Jersey’s Gov. Chris Christie’s has record low job and personal approval ratings.

While former Texas Gov. Rick Perry was the first choice of former Republican National Committee General Counsel David A. Norcross, Mr. Norcross thinks “Newt’s smarts are better in a campaign role. If Trump will listen, Newt has much to offer.”

Mr. Perry all but eliminated himself from the No. 2 slot when, during the primary campaign, he called Mr. Trump “a cancer on conservatism.”

According to Robert Schaldler, senior fellow in public piplomacy at the American Foreign Policy Council, former CIA director and Defense Secretary Robert Gates “would do the most for Trump” from the point of view of someone who knows Washington, “but Gates may not want the job.”

The consensus appears to be the notion that experience matters most.

“Gingrich is the one who has excellent D.C. experience, while Christie does not,” said Mr. Schaldler. “Newt’s personality is different from Trump‘s, and I’ve had occasion to be impressed with his intelligence and overall policy judgment.”

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