Republican presidential candidates Donald Trump celebrates his win in the New York primary on Tuesday night with family and friends at Trump Tower. (Jabin Botsford/The Washington Post)

As the Republican presidential race heads toward a possible contested convention, the three remaining candidates have begun considering vice-presidential picks — with a much different set of calculations from a normal political year.

Separate but simultaneous vetting processes are underway. Advisers to Sen. Ted Cruz of Texas and Ohio Gov. John Kasich said they are developing a list of prospects and researching their backgrounds. They are leaving open the possibility of announcing running mates as early as June, when primary voting concludes, or waiting until July at the Republican National Convention.

Front-runner Donald Trump said he is giving the choice of a running mate serious thought, but his campaign is holding off on vetting candidates until the nomination battle is over. “I would not choose a vice president unless I know I have won,” Trump said in a recent interview.

A contested convention in Cleveland would mean that instead of focusing on a running mate that would most improve their chances of swaying voters nationwide in November, the candidates may first consider whether someone will help them win over delegates.

The candidates must also game out the timing. Do they name someone ahead of the convention, as Ronald Reagan did with an unsuccessful result in 1976? Or do they wait until they get to the convention, when they could use the slot to appease potential supporters — or even to entice one of their rivals onto a unity ticket?

None of the campaigns has an answer yet.

“There’s no magic formula this year,” veteran GOP strategist Ed Rollins said. “It’s kind of like computer dating gone bad.”

Trump, who said he wants to pick an experienced political leader, may calculate that he needs a bridge to mainstream Republicans who see his candidacy as radioactive.

“There are two advantages: They can help you with the system, and the politicians have been vetted,” Trump said in the interview. “That’s the biggest advantage to a politician — their whole life they’ve been vetted and you know everything, whereas if I pick some guy out of a great corporation who has done a job, you don’t know what’s going to happen.”

Republican officials were buzzing about the vice-presidential gamesmanship as they gathered here this week along the white-sand beaches of South Florida for a Republican National Committee meeting.

Cruz campaign manager Jeff Roe said Cruz instructed his top aides to begin a vetting process this spring. “You need to be ready beforehand because you never know what you’re going to need,” Roe said. “Anything can happen. But we’re going through all the work that it takes.”

Ohio Gov. John Kasich wears a pair of boxing gloves at a town hall meeting in Annapolis, Md., on Tuesday. (Marvin Joseph/The Washington Post)

The Trump campaign is taking a different approach. Paul Manafort, Trump’s newly empowered top strategist, suggested the Cruz and Kasich campaigns should send their research dossiers to Trump Tower.

“The other two campaigns have to [start the vetting process] because they’re not relevant, they’re not credible to be the nominee,” Manafort said. “They have to do things that will make people assume they’ll be the nominee. We’re doing things that will make us the nominee.”

Ben Carson, the retired neurosurgeon who became a prominent Trump surrogate after dropping out of the GOP race in March, said it is imperative that Trump’s pick share the businessman’s worldview.

“It’s about more than politics,” Carson said. “It’s about the vice presidency, so that if something were to happen, that person could continue on and advocate for your policies.”

Carson said he does not want to be considered and would not submit to vetting. “I’ve already been vetted, more than anybody in the history of the world,” he chuckled, referring to his presidential bid. “But, no, I don’t want to be vice president.”

Seasoned Republicans are warning against selecting a running mate before the convention is gaveled in, arguing that for both political leverage and party unity the pick may need to be brokered at Quicken Loans Arena. This harks to an earlier time, such as 1940 and 1968, when GOP vice-presidential nominees were chosen at the convention to satisfy warring coalitions.

“We may be returning to a long, bygone era where there will probably be little vetting because the nominee may need to pick someone at the convention, cutting a deal as they navigate the dynamics of the moment,” said Republican strategist Karl Rove.

The intense media environment of modern politics has spawned a more thorough, lawyerly vetting process, which is what the Cruz and Kasich campaigns have begun.

In the whirlwind atmosphere of a contested convention, there could be a temptation to throw a Hail Mary. But A.B. Culvahouse, a longtime Republican lawyer who oversaw the vetting for John McCain’s 2008 selection of Sarah Palin, cautioned against sudden decisions. He said a vetting process should take at least six weeks.

“The hardest thing about the vetting process is the tension between a time-consuming process and late-emerging political considerations where, for example, an adviser or pollster says if you pick this individual then you can win this state,” Culvahouse said.

Cruz’s campaign is running its vetting in-house and is starting to scour public records, including campaign finance reports, speeches, online postings and social media. Some Cruz allies already are suggesting names, with one mentioning female senators such as Joni Ernst of Iowa and Deb Fischer of Nebraska.

The next step will be to ask potential candidates to participate in what’s known as a “full vet,” including providing tax returns and medical records and answering a detailed questionnaire.

Cruz and his advisers are preparing for many contingencies. If, for example, Trump inches towards the 1,237-delegate threshold in June, Cruz could select a running mate to attempt to alter the psychology of the race.

Similarly, Kasich instructed his top aides to begin a vetting process about two weeks ago, advisers said, and is approaching the project as an NFL team would approach a draft: preparing for numerous scenarios that might require different kinds of players.

One challenge for Kasich, however, is persuading people to submit to a full vet given the long odds he faces of becoming the nominee.

“It would be tricky for anybody because they’re not agreeing to go on the ticket with the de-facto nominee, but with an open question about whether their running mate is going to be the nominee,” said Charlie Black, an experienced GOP consultant counseling Kasich on convention matters.

Two people likely to be on Kasich’s short list are Sen. Marco Rubio of Florida and Wisconsin Gov. Scott Walker, according to one Kasich adviser, who spoke on the condition of anonymity to speak candidly.

Rubio could be on other lists as well. Since mocking him as “Little Marco” when they ran against each other, Trump suggested in a recent USA Today interview that Rubio, as well as Walker and Kasich, could have a place in his Cabinet.

Former Puerto Rico governor Luis Fortuño, a Washington GOP establishment insider and Rubio supporter, said Rubio would complement Trump.

“I think Rubio would be a tremendous addition to any ticket,” Fortuño said. “He’s still popular in Florida, he can communicate extremely well with the fiscally conservative base of the party and he can also do a great job with Hispanics.”

RNC Chairman Reince Priebus said in an interview that an open convention could affect the timing of a vice-presidential nomination. He said he would be open to delaying formal proceedings to afford the presidential nominee time to negotiate or make a decision. Even with the prospect of a somewhat chaotic convention, Priebus said he doubted that the choice would be left to the delegates.

“I think that the nominee will choose the vice president,” Priebus said. “The delegates will probably honor that choice.”

For Trump, the most appealing choices may be Rubio or Kasich because they bring delegates with them and have influence over the large, powerful delegations of their home states.

“You could see Trump, if he were stuck at 1,180 delegates, going to Kasich,” said William Kristol, editor of the Weekly Standard, a conservative magazine.

Kristol drew a historical parallel to 1976, when Reagan made the stunning decision to tap Richard Schweiker, a moderate senator from Pennsylvania, before the Kansas City convention as a means of shaking up the race. Reagan lost the nomination fight to then-President Gerald Ford.

But first Trump would have to persuade Kasich to be considered. The Ohio governor has been adamant in his refusal to be a No. 2 in anyone’s administration, a sentiment one of his advisers reinforced Thursday.

“Kasich will not participate in any vets,” the adviser said. “If somebody wants to Google him, fine, but he’s not participating under any circumstances.”