In an unbelievable contest for the White House, a credible running mate might actually make a difference.

As an apparent gap opens in public opinion polling gauging the chances that Hillary Clinton or Donald Trump can claim the presidency in November, each stands to bolster that bid with a bona fide candidate for vice president. The formula involves picking mates who can assuage the doubts that millions of voters hold for two of the least personally appealing presidential nominees in modern times.

The role of the running mate itself may be overrated in its drawing power for any presidential ticket. In fact it’s “much more likely to cost votes than gain them,’’ says Mike Murphy, a Republican media strategist who advised Sen. John McCain’s first White House bid and ran the Super-PAC backing Jeb Bush this year.

The VP pick, Murphy says in offering some pointers here about the choices facing Trump and Clinton, “is a message about the candidate making the choice.’’

For Trump, trailing Clinton by six percentage points in the RealClearPolitics average of public polls taken this month, the selection is a spotlight on executive judgment — a question heightened by his campaign’s ouster of its own manager on Monday and the defections of several Republican leaders withholding support.

It’s standard for a presidential contender to describe the search for a running mate as a quest, foremost, for someone capable of serving as president if necessary. This time, that may well overshadow any traditional consideration of geographic, political or ethnic balance — and, now in 2016, gender balance.

Clinton confronts questions of integrity ranging from her use of private emails as secretary of state to possible conflicts in the work of the Obama administration and the fundraising of her family’s foundation. Her likely opponent is pounding a social-media drumbeat of “Crooked Hillary,’’ even trying to hold her accountable for her husband’s sexual transgressions. She may well benefit from a running mate with sterling ethical credentials, perhaps even experience across bipartisan lines.

“She should pick someone that sends a strong signal to GOP moderates and independent ticket-splitters that the Bernie Sanders wild-lefty era of the Democratic primary ended with the primaries,’’ Murphy suggests.

Trump arrives as his party’s presumptive nominee with a record of reality TV shows, real estate development and casino bankruptcies. The billionaire without a day of experience in public office has built a campaign on bombast, personal assaults and promises to bar Muslims and Mexicans from the United States. Rivals within his own party and now Clinton are casting him as “unfit” for the presidency, lacking the “temperament” for commander-in-chief.

The last three successful running mates have met the presidential test: If Bill Clinton and Al Gore defied conventional odds as two young Southerners on the same ticket, Gore secured his credentials with a majority of the popular vote running on his own in 2000. Dick Cheney’s nationwide search for Dick Cheney delivered perhaps the most powerful vice president in history. President Barack Obama secured a mate possessing what he lacked: Decades of Senate and foreign policy experience. Joe Biden might have run this time, if Hillary Clinton hadn’t.

In contrast, McCain left himself vulnerable to serious questions of his own judgment and wisdom in his 2008 nomination for the White House, selecting a “pit-bull’’ of a running mate whose foreign policy expertise lay in her stated ability to view Russia from her home state of Alaska.

The attack-dog model is readily available to both Clinton and Trump, potentially enabling the nominees to rise above the fray.

For Clinton, that could mean Massachusetts Sen. Elizabeth Warren, who could rally Sanders’ dejected primary followers and happily attack Trump “all day’’ long. For Trump, that’s former House Speaker Newt Gingrich or New Jersey Gov. Chris Christie. “In a rational world, she’d be out of the race,’’ Gingrich has said of Clinton. “I’ll beat her rear end,’’ Christie said campaigning for the nomination.

Yet they pose an additional problem of polarizing voters already deeply divided over two nominees with stratospheric negative ratings. And the vaunted attack role “never works like that,’’ Murphy says, pointing to the televised presidential debates coming this fall. “You cannot bring your Veep to the big three debates.’’

If Trump wants to add credibility to his ticket, he could look to the very Congress he chastises. “I want to have somebody who can deal with Congress, who gets along with Congress, who is a Washington person,” he said at a town hall in April.

Yet Murphy, who lost his own candidate early in the GOP’s primaries to Trump’s steamrolling campaign, offers a grim prognosis for any running-mate that Trump announces this summer. “Trump will lose regardless,’’ he says, “and the best people would not accept a slot as his VP… No VP can fix Trump’s core problems: lack of judgement and character. In fact, a strong VP would only highlight Trump’s glaring weaknesses.’’

As some senators, governors and other party leaders distance themselves from Trump, someone such as Sen. Jeff Sessions, a former U.S. attorney serving in the Senate for nearly 20 years, stands nothing to lose in home-state Alabama. He is advising reluctant Republicans to get on the “Trump train.’’

In a contest in which Trump’s promises of a wall against Mexican immigrants has driven a wedge among Hispanic voters, Clinton can also reinforce her party’s claim to a fast-growing constituency that gave Obama 71 percent of its vote in 2012. This points to running mates said to be under consideration such as Labor Secretary Tom Perez, HUD Secretary Julian Castro or California Rep. Xavier Becerra.

“Trump has done so much to push Hispanics away that I’m sure the (Clinton) campaign is thinking about a Latino running mate,’’ Murphy says.

Whomever is chosen, the two will get a debate of their own, facing off about a month before Election Day, in the Oct. 4 VP debate in Farmville, Virginia, which promises to be the best-watched sideshow of all time.