The vice-presidential search has been conducted among a very small group of Clinton intimates in deep secrecy, even as some aspects were on full and intentional display. Clinton did not conceal her consideration of Senator Elizabeth Warren, a liberal firebrand who in turn has not disguised the appeal the job holds for her. Still, Democrats close to both women, including some of Warren's own advisers, have said Warren was an unlikely choice from the start.
Kaine has been a favourite for the job for months and is the name most mentioned by Democrats as the front-runner. He and Vilsack share many professional and political attributes, notably their governing experience. Both fit Clinton's ideal of low-key, loyal effectiveness, people who know both men said. Vilsack carries the additional quality of a longstanding personal friendship with Clinton.
Two Democrats described Perez as a solid third choice, but others cautioned that he may not be in the same category as Vilsack and Kaine. Several Democrats emphasised that the fact that Kaine and Vilsack are the leading contenders does not preclude the likely reality that Clinton is still weighing her choices from a larger list of contenders.
Perez met the presumptive nominee at her Washington home last weekend, one Democrat confirmed.
Perez's emergence from a crowded back field, these Democrats said, is based largely on a warm relationship with Clinton and credentials as a liberal with strong relationships with organised labour. He is also Hispanic and has served as a Spanish-speaking surrogate for Clinton.
Also in consideration has been retired Admiral James Stavridis, a former four-star commander of US forces in Europe who has strong credentials as a national security thinker and ties to Clinton from her time as Secretary of State.
Julián Castro, the Secretary of Housing and Urban Development, has been the Hispanic candidate most frequently mentioned as a potential running mate. His stock may have fallen with findings by federal investigators that he had violated Hatch Act prohibitions on mixing partisan political activity with official government duties.
Clinton campaign chairman John Podesta is leading the search, which has intensified over the last week to include several face-to-face meetings between Clinton and candidates, including Vilsack, Perez, Warren and Colorado Governor John Hickenlooper. Meetings were held at Clinton's Washington home and at other locations.
Vilsack rose through the ranks of local government to become a well-liked governor of Iowa. He was considered as a running mate for then-Senator John Kerry in 2004 and served as head of the centrist Democratic Leadership Conference while in office. In 2008, he was a short-lived presidential candidate.
He is a latecomer to the pool of possible vice-presidential choices but his star has risen over the past two weeks, several Democrats said.
Vilsack is seen as likely to deliver Iowa, a key swing state. That gives him one potential advantage over Kaine, whose home state of Virginia is also a battleground but one many Democrats judge to be safer for Clinton this year than Iowa.
"He's not a lot of bling and glitter, he's just Iowa solid," said Bonnie Campbell, an Iowa Democratic strategist with longtime ties to both Clinton and Vilsack.
So for me there is nothing more important than my rock-solid conviction that the person I choose could literally get up one day and be the president of the United States
Vilsack also comes with a compelling personal story. He was put in an orphanage as a young boy in Pittsburgh, then adopted. His adoptive mother was an alcoholic, something he mentions frequently in addresses about opioid addiction, an issue that falls under the purview of the Department of Agriculture and for which he shares a passion with Clinton.
Vilsack is, however, far from young at 65, and a narrowed field led by Kaine and Vilsack would place two white men atop a list that has included several Hispanic candidates, one white woman and one African American man - a potentially awkward optical reality for Clinton to contend with, particularly within the diverse base of the Democratic party.
Although it is unclear whether Obama offered Clinton thoughts about Kaine, senior White House officials described the one-time Mayor of Richmond as particularly in sync with Obama on issues including criminal justice reform, the death penalty and anti-poverty efforts. Kaine was an early Obama supporter back in 2007, and he served as the chairman of the Democratic National Committee following the 2008 election.
"He is this progressive Catholic. He's like an Obama Catholic," said one senior Administration official, who was granted anonymity to speak frankly about the president. "He has thought a lot about these things and can interpret Obama."
Clinton has acknowledged that the most important criteria is "experience".
"Would this person be a good president?" Clinton said on CBS News and PBS's Charlie Rose on Tuesday. "You know, I am afflicted with the responsibility gene and I know what it's like being president. I've seen it up close, I've worked for one, I've had that experience.
"So for me there is nothing more important than my rock-solid conviction that the person I choose could literally get up one day and be the president of the United States," she added.
Prodded by Rose about candidates believed to be in the running, Clinton showered them with praise, but declined to say whether they were being considered.
When Rose noted that Kaine considered himself to be "boring," Clinton laughed and replied: "And I love that about him".
She added: "He's never lost an election. He was a world-class mayor, governor and senator and is one of the most highly respected senators I know".
As for Hickenlooper, Clinton also praised him as "first class."
Warren?
"Amazing," Clinton replied. "I mean, what she has done in a relatively few years to put the agenda of inequality front and centre is something that I think we should all be grateful for."
People close to Clinton say that political considerations will come second - if they factor in at all. After the fitness test, the decision becomes about the deeply personal consideration of who she would want to be locked into a political contract with for at least four years.