Sen. Elizabeth Warren (D-Mass.) is on Democratic presidential candidate Hillary Clinton's short list of potential vice presidential candidates. Here's what you need to know about her. (Sarah Parnass,Osman Malik/The Washington Post)

The Hillary Clinton campaign has begun checking into the positions, backgrounds and financial dealings of at least three potential vice presidential candidates, Democrats familiar with the process said Tuesday: Housing and Urban Development Secretary Julián Castro, Sen. Elizabeth Warren of Massachusetts and Sen. Timothy M. Kaine of Virginia.

Clinton has also begun to winnow a list of more than a dozen potential choices, another senior Democrat said.

She is “beginning the process of narrowing a list of qualified candidates,” that Democrat said, but is still expected to consider numerous candidates.

Clinton herself has said only that her top priority is choosing someone who could become president in a heartbeat, but close allies have said she is also focused on picking a partner with whom she is personally comfortable and someone able to rally congressional Democrats and energize the party.

Clinton’s campaign was mum after an Associated Press report Tuesday naming Kaine, Warren and Castro as early choices to be vetted, and campaign spokesman Brian Fallon declined to comment.

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

“Those who talk don’t know, and those who know don’t talk,” a senior Clinton official said.

The vetting process is being run by Democratic lawyer Jim Hamilton in consultation with campaign chairman John D. Podesta and outside adviser Cheryl Mills, a longtime Clinton family confidant who served as Clinton’s chief of staff at the State Department, several Democrats said.

The circle is wider than those first three names, and others will be vetted, several Democrats familiar with the process said. All those interviewed spoke on the condition of anonymity to discuss details of a closely held and ongoing process that they cautioned remains in its early stages.

Two Democrats said they do not expect Clinton to narrow her list to a couple of finalists until much closer to the Democratic nominating convention in late July. Still, it is “reasonable” to conclude that the circle is narrowing now, one senior Democrat said.

As they attended Tuesday’s Senate lunches, some of the Democrats who have been floated as potential Clinton running mates declined to talk about it. Asked whether he was being vetted, Kaine silently winked at reporters. Sen. Cory Booker (N.J.), a frequent Clinton surrogate, paused to deliver a non-answer.

“I am not talking about that, very purposefully,” he said.

Sen. Sherrod Brown (Ohio), one of the few vice presidential prospects from a swing state, was similarly quiet about the rumors.

“I’m not going to say anything, and I’m not going to speculate,” he said. “That’s the word you’re going to get: speculate.”

Clinton mentioned Brown on Tuesday while giving an economic speech in Columbus and shared a stage with him last week in Cleveland.

“I agree with Sherrod: With the right investments and a level playing field, American workers will out-hustle and out-innovate anyone in the world,” Clinton said in the Columbus speech.

Although some potential picks, including Labor Secretary Thomas Perez and House Democratic Caucus chairman, Rep. Xavier Becerra (Calif.), were not among the three people whose vetting Democrats confirmed was underway, that does not mean they might not be considered. Retired admiral Mike Mullen, a former chairman of the Joint Chiefs of Staff, was also seen as a dark horse candidate. He had been previously vetted by longtime Clinton ally Howard Wolfson when former New York mayor Michael R. Bloomberg considered an independent bid for president.

Perez declined to comment Tuesday. He is among several potential Hispanic candidates in a year in which Clinton has focused intensively on rallying Latino voters.

In an earlier interview, Becerra said that apart from his own prospects, he hopes that Clinton is not persuaded by discussion among some Democrats that the impetus to consider or nominate a Hispanic candidate is lessened because of widespread distaste for Republican opponent Donald Trump among Hispanics.

“She will decide” whom to pick, Becerra said. “I trust her choices there. I would hope that we wouldn’t take the Latino community for granted. When you do that, you find you are surprised for the wrong reasons.”

Kaine, who became a fluent Spanish speaker as a young missionary, is seen as a safe pick who would bring support from a swing state, executive experience from his term as Virginia’s governor, and some foreign policy credibility with the left for his efforts to pass a new Authorization for Use of Military Force to limit the war on terrorism.

Yet despite a high-profile stint as chairman of the Democratic National Committee, Kaine remains less known than some of the other widely discussed candidates. He is also viewed warily by the party’s increasingly powerful progressive wing. In a CNN interview on Tuesday, Kaine won praise from Virginia’s senior senator, Mark R. Warner, who then said he hoped to build “a bipartisan coalition on debt and deficits” in the Senate. A spokesman for Sanders highlighted the exchange, which progressives interpreted as a hint that Kaine would move Clinton to the center.

Warren, an icon of the progressive wing, remains the most sought-after Democratic surrogate. On Tuesday, she starred in a Facebook video ad produced by MoveOn.org. In less than a day, Warren’s wonkish explanation of how Trump’s taxes might reveal deceptive business practices scored 2.2 million views.

In a poll last week conducted by Selzer & Co. for Bloomberg Politics, just 5 percent of Clinton supporters said Kaine would be the best possible pick. Six percent suggested Brown, 12 percent said Castro, 17 percent said Booker and 35 percent said Warren.

The idea of Brown, Booker or Warren on the ticket worries Democrats who are counting the seats needed to wrest back control of the Senate. All three, if elected vice president, would be replaced by the choices of Republican governors. In the cases of Booker and Warren, that would trigger a mid-2017 special election; in Brown’s case, it might give a Republican two full years of incumbency to build support before the 2018 midterm elections. The Democratic Senatorial Campaign Committee is already bracing for tough races that year, when seats in red states including Indiana, Missouri and North Dakota will be on the ballot.

“I’m a big Sherrod Brown fan,” said DSCC Chairman Jon Tester to Public Radio International reporter Todd Zwillich last week, “but there’s a Republican governor in the state of Ohio. It wouldn’t be good for the body” to lose him.

Asked about Warren, Tester had another, less politic concern.

“Is the country ready for two women?” he asked. “I don’t know.”

Ed O’Keefe and Karen Tumulty contributed to this report.

Correction: Longtime Clinton ally Howard Wolfson was misidentified in an earlier version of this article as Harold Wolfson.