“This party is part of her DNA and she deserves the respect,” said Minyon Moore, a strategist who has advised Mrs. Clinton for decades. “How many candidates have lost their campaign? Some twice, and they’re still running again, potentially. She has a stature in this party that she has earned.”

At a time when Democrats are driven by their desire to defeat Mr. Trump, Mrs. Clinton knows the strengths and weaknesses of the crowded primary field more intimately than perhaps anyone. Over the past quarter-century or so, she has served as a mentor, colleague, supporter or rival to nearly every Democrat running or considering a run. She vetted at least five of the current or possible 2020 candidates as her running mate in 2016, going so far as to print signs for one of them, Senator Cory Booker of New Jersey. And her campaign alumni pepper the upper reaches of numerous 2020 campaigns.

In recent months, a parade of ambitious Democrats has held private sessions with Mrs. Clinton, who has counseled them about the unmatched rigors of the campaign trail and hardships of facing Mr. Trump, according to people familiar with the conversations. Beyond Mr. Biden and Ms. Klobuchar, she has spoken with Senator Kamala Harris of California; Mr. Booker; Senator Elizabeth Warren of Massachusetts; John Hickenlooper, the former governor of Colorado; and the former housing secretary Julián Castro, among others. She and Mr. McAuliffe speak regularly. Gov. Steve Bullock of Montana is trying to schedule a meeting with her.

“She knows almost everybody who’s running, and with most of the people she has a close relationship,” said Mr. Hickenlooper, who interviewed to be Mrs. Clinton’s running mate in 2016 and had dinner with her last year, as he prepared for a possible presidential bid. “You talk about divided affections. She’s got it.”

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Not everyone has sought Mrs. Clinton’s advice. Senator Bernie Sanders of Vermont, whose 2016 primary challenge ruptured their relationship, has not spoken with Mrs. Clinton, or even reached out, despite the imperative to make inroads into her coalition. Beto O’Rourke, the former Texas congressman considering a run, has not spoken with Mrs. Clinton either.

A notable name not to consult Mrs. Clinton is Senator Kirsten Gillibrand of New York, whose once close relationship with the Clintons deteriorated after she said President Bill Clinton should have resigned over his sexual relationship with a White House intern.