Big-name Republicans aren't exactly jostling each other for the chance to be Donald Trump's apprentice.

Democrats are watching as Hillary Clinton goes on the road Monday in pivotal Ohio with Sen. Elizabeth Warren (D., Mass.), the progressive icon. It sure seems like a tryout.

After a bruising primary fight with Bernie Sanders, Clinton does need help with her party's disappointed left and young voters, analysts point out. And Trump, after two bumpy months as presumptive GOP nominee in which he has wallowed in gaffes and failed to raise much money, could use political and policy heft in a running mate - somebody soothing to anxious party leaders and donors.

So bring on the vetting, and the speculation. With less than a month until the Republican convention begins in Cleveland, followed immediately by the Democratic conclave in Philadelphia, the season is nigh to float names and shoot them down.

Factors in a presidential candidate's choice of running mate include qualifications and personal chemistry, but most have aimed for balance - ideological, demographic, geographic - or picked someone who addresses a perceived weakness.

Thus, in 1980, Ronald Reagan picked George H.W. Bush, who was more moderate politically and compensated for the former California governor's lack of foreign policy experience. Eight years ago, Barack Obama, then a young first-term senator, chose the ultimate Washington insider, Joe Biden.

Clinton's campaign has begun deep dives into the positions, backgrounds, and finances of at least three potential vice presidential candidates, according to Democratic sources: Warren, Sen. Tim Kaine of Virginia, and Obama's Housing and Urban Development secretary, Julian Castro, as the Associated Press first reported last week. Insiders say the process is early, others are under consideration, and the number subject to vetting is likely to grow.

Among the other names mentioned as possible Clinton running mates: New Jersey Sen. Cory Booker, Sen. Sherrod Brown of Ohio, and U.S. Rep. Xavier Bacerra of California.

A twist in an election year full of strange twists: both presumptive nominees face high "negative" ratings and may want running mates who are, well, better liked.

As Trump faces his choice, the striking thing is the number of Republicans who would make fine political sense as running mates have slammed the door on being considered.

Usually there is a long line of VP aspirants, from Capitol Hill, statehouses, major cities, and sometimes, retired military commanders - all putting their names out there, discreetly, of course, and often through surrogates.

Yet Ohio Gov. John Kasich and Florida Sen. Marco Rubio - one-time competitors for the GOP nomination who represent major swing states - have made their disinterest clear, as has Robert M. Gates, the former defense secretary.

Other rising party stars also have said they would not run with Trump, including Govs. Nikki Haley of South Carolina and Susana Martinez of New Mexico, whom Trump attacked as incompetent while campaigning there this month.

"Most able and upwardly mobile politicians are interested in being VP, but this year there seems to be a big exception on the Republican side," said Joel Goldstein, a law professor at St. Louis University who has studied the vice presidency. "The pool is shallow."

A major potential reason: Trump's explosive personality.

"If you're the running mate, your job is to be the first responder when the candidate is under fire," Goldstein said. "Most politicians across the spectrum don't agree with or want to be associated with the kind of statements Mr. Trump has made that [House] Speaker Paul Ryan calls racist."

In recent weeks, Trump has said that an Indiana-born federal judge of Mexican heritage could not be objective in presiding over a fraud case involving one of his former businesses. He also proposed putting Muslim Americans under special surveillance and repeated his call for a ban on Muslim immigration to the U.S. in the wake of the Orlando terrorist attack.

"He needs some stability," Stu Spencer, a GOP strategist for half a century, told the Los Angeles Times recently. "He needs some class. He needs somebody that people trust. Anybody of that caliber who attaches themselves to Trump is insane."

Other names that have been bruited for Trump's No. 2 include Govs. Christie of New Jersey and Mary Fallin of Oklahoma, as well as Sen. Jeff Sessions of Alabama, the first senator to endorse Trump and a vigorous defender. Sens. Bob Corker of Tennessee, Tom Cotton of Arkansas, and John Thune of South Dakota have also surfaced as possibilities.

Trump has said he most wants someone who would make a "great" president in the event he or she assumed the position. He ruled out a businessman or political outsider like himself, saying he wants an understudy experienced in the ways of government.

It's not surprising to Pennsylvania Republican strategist Charlie Gerow that ambitious pols are holding back, considering the recent tribulations of the Trump campaign. But he does not expect that to last.

"The truth is a lot of people would like to be the vice presidential nominee on a major-party ticket - it's a valuable thing," Gerow said. "If the nominee for president of your party calls you to serve, it's very difficult to say no."

On the Democratic side, it's unlikely that Clinton will call on Sanders. There's no love lost between them, and the Vermont senator has made it clear he did not want or expect to be seriously considered.

Warren, champion of tougher Wall Street regulations, has put herself forward by taking on Trump in recent weeks. She would excite the liberal voters attracted to Sanders' challenge, allies say. But Warren only endorsed Clinton after the primaries were over, the two are not close, and she has a number of other options.

Aides say Clinton likes Brown, the Ohio senator, and they appeared quite comfortable together sharing a stage during a campaign visit to Cleveland recently. He also is to Clinton's left, with a more protectionist bent on trade issues, and his popularity could help on the margins in carrying Ohio - the ur- swing state.

But if Brown is taken for the ticket, Republican Gov. Kasich would appoint a successor, which would hurt Democratic hopes of retaking the Senate.

Clinton has said she loves Booker's "charisma," and he can stir crowds. The New Jersey senator could overshadow Clinton, a more pedestrian campaigner, and he also shares her deep ties to Wall Street, a major knock against the presumptive nominee in the liberal wing of the party.

Picking Castro, the former mayor of San Antonio, could fire up Latino voters and help weld that important bloc to the Democratic Party for a generation. He's 41, and a strong communicator. On the other hand, does she need a boost with Latinos while running against Trump? Castro also is not experienced on the national stage.

Kaine, a former Richmond mayor, Virginia governor, and Democratic National Committee chairman, endorsed Clinton early in the 2016 cycle. He represents an important swing state that is tilting Democratic. Kaine also has been vetted before, reportedly as a runner-up to Biden in 2008, so he's a safe choice in that respect. A centrist Southerner, Kaine might help with Clinton's deficit among white men, but some say he may not excite the base.

And as unusual as this race has been, exciting the base may still matter.

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@tomfitzgerald