Carson Says Trump Could Consider Democrat for Running Mate

Ben Carson, the former rival now helping presumptive Republican presidential nominee Donald Trump choose a running mate, said Thursday he doesn’t want the position and that Democrats may be among those considered.

“I’m not interested in doing that for a number of reasons,” Mr. Carson said in an interview with The Wall Street Journal. “I don’t want to be a distraction. I’m sure you remember how crazy the media was about me, I don’t want to be a distraction, it’s too important a time in our life.”

As for the wide net being cast for candidates, he added: “We would consider people who are Americans and who put America first.”

Just two days after his last two rivals quit the primary, Mr. Trump’s effort to choose a vice president is beginning to take shape. Mr. Carson is the public face on the search process, while top campaign aide Paul Manafort is maintaining tight control over the process, according to people familiar with the campaign.

Similar work is being done for likely Democratic nominee Hillary Clinton. Her search for a running mate is being led by John Podesta, her campaign’s chairman. Mrs. Clinton hasn’t revealed details about specific people or attributes she will consider.

“The most important quality is that this person could become president literally on a moment’s notice,” Mrs. Clinton told CNN Wednesday. “There is no more important quality than that.”

For Mr. Trump, his running-mate choice could be the most important signal that he is seeking to unify the GOP. And, given that Republican National Convention delegates aren’t bound to abide by his choice, the presumptive nominee could face an arena of delegates rebelling against his running-mate choice if he sends the wrong signal.

“He could get to Cleveland and designate someone as his vice president and have the convention lose its mind,” said Erick Erickson, an anti-Trump conservative activist.

“Three weeks ago, I was on a call with some conservative activists and this topic came up,” Mr. Erickson said. “The consensus was that if you couldn’t get Ted Cruz or Marco Rubio on the ticket, that they would give a Shermanesque refusal to Trump’s pick,” he added in a reference to two of Mr. Trump’s former presidential rivals.

Such a plan would almost certainly have to be led, publicly, by Mr. Cruz, who has so far shown no inclination to do so. His campaign aides didn’t respond to requests for comment Thursday.

Mr. Trump’s campaign manager, Corey Lewandowski, said it is implausible that Mr. Trump’s selection won’t be nominated as his running mate. “I can’t imagine any scenario where the GOP nominee doesn’t have the opportunity to pick his own vice-presidential candidate,” he said.

But such a scenario nearly did play out in 2008, when then-GOP nominee John McCain was considering tapping Sen. Joe Lieberman, a pro-choice former Democrat, to be his running mate. The campaign reached out to people influential with convention delegates and concluded that nominating Mr. Lieberman would tear the party apart, McCain aides said.

“In ’08 we actually did a lot of soundings about Lieberman,” said former McCain aide Charlie Black. “Remember, you were asking a Republican convention to nominate a Democrat. We could have gotten Joe through the convention but we would have had a split party afterward.”

For now, Trump aides are assembling a broad roster of candidates, said Mr. Lewandowski, who added that the New York businessman will “in the near future” reveal who is on the campaign’s vice-presidential search committee beyond Mr. Carson.

Mr. Trump on Thursday said there is “probably a 40% chance” he would pick one of his 16 dispatched GOP presidential rivals but that he was “unlikely” to choose the last man standing, Ohio Gov. John Kasich who suspended his campaign on Wednesday, a day after Mr. Cruz dropped out of the primary.

“I’ve gotten to be friends with a lot of those people, and I guess perhaps enemies with a couple,” he said in an interview on CNBC.

Mr. Trump said he was looking for someone with political experience because he already knows the business world. He also said rapport was “awfully important,” noting the affable relationship between President Barack Obama and Vice President Joe Biden.

“I think having somebody that can get legislation through and help me with that would be good,” Mr. Trump told CNBC. “Of course, always the first reason is if something should happen, somebody that can serve and serve well and be a great president. And that’s always—you always start with that. And after that, it’s really a question of rapport. I think rapport is very important.”

Messrs. Carson and Lewandowski declined to suggest details of the campaign’s vetting process or suggest names of potential running mates for Mr. Trump. Some potential candidates have been quick to rule themselves out.

On Thursday a top aide to Ohio Sen. Rob Portman said he won’t submit to being vetted for consideration. “It’s not happening, period,” Mr. Portman’s campaign manager said. South Carolina Gov. Nikki Haley said Wednesday that she won’t be in the running either.

But in Maine, Gov. Paul LePage said at a Wednesday night town meeting that he is considering a 2018 Senate run “if I’m not into the Trump administration.” And former House Speaker Newt Gingrich said by email Thursday that he “might” be willing to serve as Mr. Trump’s running mate, though he otherwise declined to discuss his interest.

Write to Reid J. Epstein at reid.epstein@wsj.com and Beth Reinhard at beth.reinhard@wsj.com