Sen. Elizabeth Warren is gaining on Joseph R. Biden in the Democratic presidential race and, of course, she has a plan to beat him: Shatter the image of the former vice president as heir to the Obama legacy.

The senator from Massachusetts is offering Democrats an Obama-esque vision that Mr. Biden can’t match with his meandering message about fighting racism and restoring the status quo in a post-Trump America.

Although Ms. Warren would steer the country much further left, her plans for a socialist-style makeover echo Mr. Obama’s promise in 2008 of “fundamentally transforming the United States of America.”

Neil Sroka, a spokesman for Democracy for America who worked on the 2008 Obama campaign, said Ms. Warren is a more natural heir to the Obama “change” mantle.

“I think that in some ways — both explicitly and implicitly — her candidacy is much more clearly about change than Joe Biden’s could almost ever be, because she is not only exclusively calling for change in her policy proposals, but because she isn’t tied to a past administration as clearly as Biden is,” Mr. Sroka said.

He said Hillary Clinton faced a similar challenge with Barack Obama in the 2008 presidential primary race.

“She understood the political winds of the moment and would try to have a change message, and she certainly would have been a change compared to Bush, but just because of who she was and her history within the party and within national politics, she struggled,” he said. “Biden has a similar problem.”

Mr. Biden held on to the front-runner position, but his poll numbers have been on a downward trajectory since he entered the race in April while Ms. Warren’s have been climbing.

National and Iowa polls this week show the race has narrowed to a two-way contest between Mr. Biden and Ms. Warren.

At a Monday rally in New York’s Washington Square Park that drew more than 20,000 people, Ms. Warren made the pitch that she is the one to move the Democratic Party forward.

“There’s a lot at stake in this election, and I know people are scared, but we can’t choose a candidate we don’t believe in just because we’re too scared to do anything else. And Democrats can’t win if we’re scared and looking backward,” she told the crowd.

She insisted later that she wasn’t talking about Mr. Biden, who has pinned his White House ambitions on his ties to the popular Mr. Obama and a promise to make American normal again.

“It’s talking about whether we’re going to turn backwards and just say, ‘The only problem is Trump. If we get rid of Trump, everything is going to be just fine,’ ” she said on MSNBC’s “The Rachel Maddow Show.”

Ms. Warren’s allies, though, are concerned that she could be peaking too soon.

They point out that massive crowds rallied behind Howard Dean’s change message in 2004 before his presidential campaign fizzled out after a lackluster showing in Iowa.

Others see plenty of room for Ms. Warren to keep expanding her base.

“The two biggest honey pots for Elizabeth Warren to grow are Biden supporters and undecided voters, all of whom are electability voters who increasingly see her as the best foot forward against Donald Trump,” said Adam Green, co-founder of the Progressive Change Campaign Committee.

He said part of Ms. Warren’s appeal is her ability to deliver the Obama message in new ways.

“Warren is merging the desire for big change that Barack Obama campaigned on with Donald Trump’s strategy of uniting the bases of both parties against the corrupt system itself,” Mr. Green said. “Asking whose side the government is on is not a left-versus-right issue. It is an insider-versus-outsider message. That is the appeal of her message. … That’s why we cannot afford to nominate an establishment insider.”

The pro-Warren Progressive Change Campaign Committee has tracked voters who have switched support from Mr. Biden to Ms. Warren. Their comments highlight Ms. Warren’s image as a transformational figure.

A voter identified as David M. of Tampa, Florida, told the group that he made the switch because “Biden seems old-school political with no concrete values” and Ms. Warren offers a “realistic pathway to a modern, fair economy.”

A modern, fair economy was precisely what Mr. Obama promised in 2008.

“You can choose policies that invest in our middle class and create new jobs and grow this economy so that everyone has a chance to succeed, not just the CEO, but the secretary and janitor, not just the factory owner, but the men and women on the factory floor,” Mr. Obama intoned at a 2008 campaign stop in Columbia, Missouri.

The stories of voters swinging their support behind Ms. Warren back up what activists see in Iowa.

“I haven’t heard anyone say they are moving to Biden,” said Rod Sullivan, a supervisor of Johnson County Iowa, and a Warren supporter.

Mr. Sullivan backed Mr. Obama early on in the 2008 presidential race and says he sees similarities to Ms. Warren.

“They are both beneficiaries of that hunger and thirst for significant change and … in two different ways they are both groundbreaking,” he said.

But Democratic strategist Zach Friend, who worked on the 2008 Obama campaign, scoffed at the idea that Ms. Warren was closer to the Obama mold.

“There may not be a candidate that is the direct heir to the Obama legacy, but it’s clear some candidates, like Vice President Biden, are embracing Obama policies more than others — including Sen. Warren and [Sen. Bernard] Sanders,” he said. “Obama remains one of the most popular figures with Democratic voters, so running away from his policies, or questioning some of his approaches, might not be the best strategic decision and for that matter might not line up with the average Democratic voter’s priorities.”

The Biden camp has pushed back against calls by Ms. Warren and Mr. Sanders to scrap Obamacare and replace it with a “Medicare for All” government-run health care system.

Mr. Biden’s team argues that such socialist-style policies are unpopular with voters in Rust Belt states such as Pennsylvania, Michigan and Wisconsin that Democrats need to win back in 2020.

Mr. Sullivan said Mr. Obama’s real gift was his personal touch with voters and that Ms. Warren is closer to him in that regard on the campaign trail.

“Biden is leaning on the Obama policy stuff, which I don’t think Democrats dislike, but it didn’t light them on fire,” he said. “The stuff that lit them on fire with Barack were personal characteristics.

“Obama really made you feel like you were being heard. I don’t think Joe has that,” he said. “And Warren, she is a lot better in that area than people thought. She is a good listener, and her mind is a steel trap. She remembers everything.”

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