Clinton spent a rare full day off the campaign trail on Thursday at her home in Washington, where she huddled with a team of advisers and lawyers tasked with helping her search for a vice presidential running mate.
For Trump, the process remains fluid, according to media reports, but three people have emerged as the most likely options after two prominent senators bowed out of consideration this week.
Trump is likely to announce his selection toward the end of next week in a bid to ramp up excitement for the GOP convention, which opens in Cleveland on July 18.
Clinton, CNN reported, is most likely to announce her selection the morning after the GOP convention concludes in an effort to drown out the positive news that generally follows a candidate's prime time national address accepting his party's presidential nomination and turn the focus to her own convention the following week in Philadelphia.
With just two weeks to go, here is a look at where things stand in the quadrennial political parlor game known as the veepstakes.
Virginia's Kaine emerges as Dem front-runner
Kaine hails from a swing state where he has won statewide election twice, once as governor and then as senator.
Kaine brings a moderate Senate voting record and executive experience, underscoring a core Clinton message, that she is more experienced than Trump. As a former chairman of the Democratic National Committee, he also doubles down on Clinton's ties to the party establishment and appeal to registered Democrats, who were the driving force behind her clinching the nomination.
Another potential perk of a Kaine pick: He is fluent in Spanish, which could prove a valuable asset as Clinton seeks to court Latino voters in response to Trump's criticism of Mexican immigrants.
Kaine also has one thing in his favor that another candidate believed to be on Clinton's short list does not: a Democratic governor to name his replacement to the Senate. Should the Senate come down to a 50/50 split, a possibility, Clinton could regret picking a senator as her running mate if it also meant losing the Senate majority.
That could be a risk in the case of Sen. Sherrod Brown of Ohio, who has also been mentioned as a potential Clinton running mate.
Politico reports that Brown is a favored option by Clinton and would bring a more progressive voice to the ticket than Kaine, but because Ohio has a Republican governor, a Clinton-Brown victory would also mean handing a Senate seat to Republicans for at least two years until a special election in 2018.
Brown, however, has been a forceful critic of free trade agreements and analysts say he could therefor give voice to disaffected voters across the Midwest and Rust Belt and help blunt one of Trump's main arguments to working class voters in those states.
Brown also boasts strong ties to organized labor, a key subset of voters and rich source of Democratic volunteers.
Warren could provide the most help for Clinton in winning over the Democrats' liberal base after a long primary fight with Sen. Bernie Sanders, who regularly outperformed Clinton among younger, more liberal voters.
Warren did not endorse Clinton until after the primary calendar had run its course, the only female Democratic senator to do so. Since offering Clinton her backing, the two have campaigned together and Warren visited Clinton's campaign headquarters in Brooklyn to offer the staff there encouragement.
CNN reports a handful of others remain under consideration, but are considered long shots. They include two members of Obama's cabinet, Agriculture Secretary Tom Vilsack, a former two-term governor of Iowa, and Labor Secretary Tom Perez. Another Obama cabinet secretary, Julian Castro, whose name was mentioned early in the process, has been largely dropped from consideration, as has Rep. Xavier Becerra of California.
Trump sees potential options dwindle as choice nears
As Trump approaches the single biggest decision of his young political career, the number of options before him has decreased in recent days, while other candidates are raising their profiles in advance of the selection.
Two senators mentioned as potential options for Trump, Bob Corker of Tennessee and Jodi Ernst of Iowa, have both told the presumptive Republican nominee in recent days they are not interested in assuming the role of vice president.
Ernst, a first-term Republican, met with Trump on Monday and he offered praise for her, promising he will "see her again," Politico reported. But in a subsequent interview with the news outlet, Ernst said she told Trump she would like to continue her work as senator rather than join the ticket.
"I made that very clear to him that I'm focused on Iowa. I feel that I have a lot more to do in the United States Senate. And Iowa is where my heart is," Ernst told Politico. "I'm just getting started here. I have a great partner with Chuck Grassley, we've been able to accomplish a lot. And I think that President Trump will need some great assistance in the United States Senate and I can provide that."
Corker, chairman of the Senate Foreign Relations Committee, was said to have been "very seriously" vetted by Trump, who has said he would like a running mate who has experience with the legislative process on Capitol Hill.
Corker offered praise for Trump and said he would like to be considered for a cabinet position, perhaps as secretary of state, if Trump becomes president. But, he told The Washington Post he was not interested in the job of vice president.
"There are people far more suited for being a candidate for vice president, and I think I'm far more suited for other types of things," Corker said.
The departures leave a handful of names remaining atop Trump's list, according to NBC News.
Pence is a former six-term congressman who won election as governor four years ago. He brings a conservative record both on fiscal and social issues to the campaign, which could help allay concerns among some on the right about some of Trump's positions that are at odds with conservative orthodoxy.
Pence made national headlines last year for signing a controversial "religious freedom" bill that prohibits business owners from facing legal consequences for denying customers service on religious grounds. Critics said the bill legalized discrimination against gays and lesbians in the wake of the Supreme Court's ruling legalizing same-sex marriage.
Former House Speaker Newt Gingrich, who has praised Trump as a candidate on several occasions, has also reportedly been under consideration. Gingrich told Fox News on Thursday he would feel "compelled to serve the country" if Trump asked him to be his running mate.
Gingrich appeared alongside Trump at a rally in Cincinnati on Wednesday, where he praised Trump's campaign as "historic."
Previously, however, Gingrich went on record calling Trump's criticism of a federal judge with Mexican heritage "inexcusable."
Another sitting governor, Chris Christie of New Jersey, is also widely believed to be on Trump's short list.
Christie became an early Trump supporter after ending his own presidential bid after disappointing finishes in Iowa and New Hampshire.
Christie has appeared at Trump's side at various points in the campaign and has served as a regular behind-the-scenes confidant, fundraiser and messenger between Trump and establishment Republicans.
NJ.com cited multiple anonymous sources close to Christie and Trump, saying it was unlikely Christie would be Trump's ultimate choice, but that he would continue to serve as a top surrogate and adviser through the general election.