Sen. Timothy M. Kaine (D-Va.), left, and Agriculture Secretary Tom Vilsack, right, have emerged as the leading contenders to be Hillary Clinton’s running mate. Democrats close to Labor Secretary Tom Perez, center, do not expect him to get the nod. (From left to right: Jahi Chikwendiu/The Washington Post; Bill O'Leary/The Washington Post; Jeffrey MacMillan)

Once again, Hispanics are in the mix of potential vice-presidential candidates. And once again, they appear poised to be passed over for a white guy.

It has happened before — to Henry Cisneros in 1984 and Bill Richardson several times. This year, two Hispanic men, Housing Secretary Julián Castro and Labor Secretary Thomas E. Perez, have been among the Democrats Hillary Clinton is considering to pick as a running mate.

The 2016 presidential campaign cycle began with the predominant theory that Clinton would need to seriously consider a Latino for the No. 2 slot, given the rapid growth of Latinos and campaigns by two Latino Republican presidential candidates.

That is no longer the case.

First, Republican Donald Trump’s historic unpopularity with Latino voters may have erased any political urgency to choose a Hispanic candidate. Second, both leading Hispanic contenders lack experience that Clinton has told friends and advisers she considers critical for the role: Neither has military or national security experience — considerable drawbacks in a time of heightened domestic and global strife.

Housing and Urban Development Secretary Julián Castro is on Democratic presidential candidate Hillary Clinton's short list of potential vice presidential candidates. Here's what you need to know about him. (Sarah Parnass,Osman Malik/The Washington Post)

In the waning days of the Clinton “veepstakes,” Virginia Sen. Timothy M. Kaine and Agriculture Secretary Tom Vilsack have emerged as her leading choices, according to several Democrats familiar with her thinking. But many of those Democrats say other contenders remain in the mix, with Perez mentioned most frequently as an additional likely finalist.

Democrats also say that the potentially historic selection of a Hispanic running mate sparked months of bitter, backroom jockeying between the close aides and associates of Castro and Perez.

Perez boosters suggest Castro’s very public profile and early mention as a possible Clinton running mate may have hurt him in the end. Both endured embarrassing headlines this week, prompting loyalists to privately accuse the opposing side of planting the stories.

All of it has left Latino leaders in business, media and politics anguished, conceding that Clinton faces no significant risk of losing Latino support if she skips over Castro and Perez.

Richardson — the former New Mexico governor who was considered for vice president in 2000, 2004 and 2008 — is a loyal Clinton supporter but recalled the raw political nature of the vice-presidential search process for Al Gore, John F. Kerry and Barack Obama.

“At the very end, what makes the final decision is a bunch of white guys that are polling. They are not, in my judgment, sensitive about national security or ethnicity,” he said. “They are looking for additional votes for the presidential candidate.”

We narrowed Clinton’s vice presidential possibilities to 27. Now you pick one.

When he was told he did not get the job in 2004, “Kerry said something like, ‘Well, we have a shot at North Carolina,’ ” Richardson recalled. In 2008, he was told by Obama campaign staffers that he did not poll well in the Hispanic community because of his last name.

The fact that Richardson was not picked, however, could have had as much to do with reservations about him by the nominees as with the electoral college priorities of their campaigns.

Cisneros, who served as mayor of San Antonio, did not respond to a request for comment. Democratic nominee Walter F. Mondale’s decision to interview Cisneros in ’84 electrified Latinos, and it was widely read as an indicator of their growing political clout. But privately, Cisneros previously recalled, Mondale told him it was “highly unlikely that, as a mayor, I would be selected.”

Alex Nogales, president and chief executive of the National Hispanic Media Coalition, has toiled for years with top Hollywood bosses and network news executives about the lack of representation on television news and the big screen. He said Castro or Perez would make Hispanics “a visible part of the cultural life of the United States.”

“All of a sudden, this community would have more of a flexibility to discuss its problems, to discuss its good points, its qualities, its humility, its hard-working nature,” he said. “All of these attributes would come to the fore. Would it help the Latino community? A thousand times over, yes.”

Javier Palomarez, president of the U.S. Hispanic Chamber of Commerce, said he has spoken directly with Clinton about Castro’s prospects on two occasions in the past year. He has also conveyed his preference to Clinton’s political director, Amanda Renteria, the first Hispanic to hold the senior campaign position. When Palomarez first spoke with Clinton, he said that she agreed that Castro deserved serious consideration.

But when Palomarez raised the issue with her again during a telephone call in early May, “she seemed to wane just a little bit,” he recalled. When he mentioned Castro, Clinton said, “Whatever role he has, he will have an important role in my administration,” Palomarez recounted.

People close to Perez said they do not expect him to get the nod either, based in part on his lack of foreign policy experience and what they perceive as less of an imperative on Clinton’s part to put a Hispanic in the ticket.

On Wednesday, Palomarez’s group announced its endorsement for Clinton after previously taking the un­or­tho­dox step of endorsing Castro for vice president.

“I should hope that her campaign — as white as it is — would recognize the power of the Hispanic vote and recognize that Julian Castro, beyond being Hispanic, is the right man for the job,” Palomarez said.

The push for a Hispanic vice president has been muted from the start. An attempt by Texas state lawmakers to send Clinton a letter backing Castro fizzled in recent days. Facebook groups touting Castro’s candidacy mostly lie dormant. Members of the Congressional Hispanic Caucus expressed general support for the concept of a Latino vice president but did nothing to actively campaign for Castro or Perez, partly out of respect for Rep. Xavier Becerra (D-Calif.), a senior member of the group who was also briefly touted as a possible contender.

The urgency to pick a Latino running mate faded once Trump defeated Sens. Ted Cruz (R-Tex.) and Marco Rubio (R-Fla.), as well as former Florida governor Jeb Bush, who was seen as a formidable foe by Democrats because of his multicultural and bilingual family. Trump won the GOP nomination in part by calling for a Mexican border wall, calling some Mexican immigrants criminals and rapists and mocking Bush and his Mexican-born wife for speaking Spanish.

A recent poll by Univision among Hispanic voters gave Clinton a 45-point lead over Trump.

Of the two Hispanics in the mix, Castro enjoys a wider profile, but neither is well-known by the broader Democratic Party. Both men have executive experience: Castro ran the nation’s seventh-largest city and a Cabinet department, while Perez was county council member and deputy attorney general before leading the Labor Department.

But this week, both were the subject of unfavorable news reports. On Monday, the Associated Press reported that Castro violated the federal Hatch Act barring government officials from using their official perches for political reasons. A Wall Street Journal article on Wednesday detailed how Perez’s grandfather once worked for the Dominican Republic dictator Rafael Trujillo. For decades, millions of people from across Latin America have fled their home countries to escape dictatorial regimes.

Kaine or Vilsack, meanwhile, could struggle to win over Hispanics. Both are Roman Catholic — as are most Hispanic Americans — but both could be seen as having robbed qualified Latinos of the chance to serve a heartbeat from the presidency.

As Iowa governor, Vilsack signed an English-only bill in 2002 under intense political pressure and with his reelection looming. He apologized to Hispanic organizations for the decision in 2004 when Kerry considered him for vice president, but he declined to discuss the issue with reporters when his wife, Christie Vilsack, ran for Congress in 2012 against the man who had drafted the bill as an Iowa legislator — Rep. Steve King (R-Iowa), a staunch opponent of immigration reform.

Kaine would have an easier time telling his story to Latinos. A freshman senator and former governor of Virginia, he took a break from his studies at Harvard Law School in the early 1980s to work with Jesuit missionaries running a Roman Catholic school in Honduras. He learned to speak Spanish — and speaks it frequently in public with a native-sounding accent.

At the White House, which is closely monitoring Clinton’s selection process, press secretary Josh Earnest on Wednesday offered praise for Vilsack, Perez and Kaine. He made a point of saying that Obama considered Kaine “one of his own,” even though he had not served in the Cabinet.

Juliet Eilperin, Anne Gearan, Abby Phillip, Karen Tumulty and John Wagner contributed to this report.