Elizabeth Warren is a first-term U.S. senator, a former Harvard law professor, an expert in financial regulations, and the woman some Democrats wish would run for president, but most recently she's known for something else — picking a fight with the president.

President Barack Obama and Warren are in the midst of an unusually public spat, using blunt language to call each other out and to convince their Democratic colleagues, and the American public, that they are right and the other is wrong.

The Massachusetts senator and the president are wrangling over the proposed Trans-Pacific Partnership, a massive trade pact the U.S. is negotiating with 11 other countries, including Canada. Most Republicans are on side with the deal, while many Democrats are opposed to it, putting Obama in an awkward position.

That appears to be of little concern to Warren, who is leading the charge against how the trade deal is being negotiated in secret and its potential consequences for American workers, regardless of any embarrassment that might cause Obama.

"She is fearless. She cares about this, and these topics, and doesn't care about the rest," said Mary Anne Marsh, a Boston-based Democratic strategist.

In a relatively brief period, she was elected to the Senate in 2012, she has emerged as a powerful player with widespread national appeal, as indicated by the "Draft Warren" campaigns that are encouraging her to challenge Hillary Clinton for the 2016 presidential nomination. Warren has repeatedly said she's not interested.

Playing Washington's inside game

She's managing to be a vocal critic of Obama's policies while still being taken seriously by her fellow Democrats and earning their respect. Contrast that delicate balancing act with someone like Republican Ted Cruz, whose fiery rhetoric and stunts in the Senate have turned off people in his own party and drawn accusations of grandstanding.

When Warren lobbies her colleagues on the Trans-Pacific Partnership and other issues, they listen. They find her intelligent and knowledgeable about economics. As a law professor, Warren specialized in bankruptcy, and her resumé includes establishing the Consumer Financial Protection Agency.

President Barack Obama leans in to kiss Elizabeth Warren when she was running for her Senate seat during a campaign event in Boston in 2012. (Stephan Savoia/Associated Press) It also includes leading a congressional watchdog committee that aimed to keep accountable the Wall Street firms and big banks that got bailouts after the 2008 financial crisis.

"One of the reasons she has so much influence is because she is very smart and she talks about these issues in very substantive ways," Massachusetts Democratic Representative Seth Moulton recently told the Boston Globe. "It's not just rhetoric."

As Marsh explained, Warren has also played "the inside game" in Washington. She's supported her colleagues and their causes, campaigned for them in the mid-term elections, helped raise a lot of money and made donations.

She's gained public support outside of Washington by sticking to her campaign promises, said Marsh, and people not only in Massachusetts but across the U.S. view her as someone who is fighting for them.

"She's accumulated a lot of power in a short amount of time, but she's very strategic and very disciplined about how she uses it," said Marsh.

Her battle with Obama over the Trans-Pacific Partnership has involved her lobbying colleagues to vote against a bill he wants that would give him fast-track authority to complete the deal. She has given interviews and penned op-eds where she blasts the administration for negotiating the deal in secret. Warren claims it's a rigged process that will lead to a rigged outcome.

On Monday, she published a 15-page report called Broken Promises that she said details more than two decades of failed enforcement of labour and environmental standards in past trade deals, including the one with Canada and Mexico, the North American Free Trade Agreement.

Obama calls Warren 'absolutely wrong'

The report was yet another salvo in the feud over Obama's trade agenda. But this squabble isn't the first time Warren has caused trouble for her president. Last year, she led an effort to shut down Obama's nomination of Antonio Weiss for a key Treasury Department job. She didn't like his Wall Street connections and she made her opposition known. Others joined her and the nomination was withdrawn. She's also opposed the White House over budget measures.

A Trans-Pacific deal is important to Obama and his legacy as he prepares to leave office, and he's pushing back against Warren. In an interview two weeks ago with Yahoo, he pointedly stated that the senator is "absolutely wrong" when she talks about the deal's potential to undo tighter regulations on Wall Street.

He also suggested that Warren was playing politics: "The truth of the matter is that Elizabeth is, you know, a politician like everybody else.… She's got a voice that she wants to get out there, and I understand that. On most issues, she and I deeply agree. On this one, though, her arguments don't stand the test of fact and scrutiny."

Ouch. Since then, Obama, the White House and Warren have had to answer repeated questions from reporters about the strained relationship. Obama also had to fend off an accusation from one senator who called the comment sexist, illustrating how careful the president has to be when debating Warren.

Both Obama and Warren say their dispute is nothing personal. The optics, however, aren't ideal. What you have is a Democratic president urging Americans to ignore an influential Democratic senator and her allies on the left when she talks about the trade deal, and a popular Democrat urging Americans and Congress not to believe what the president says when he touts the benefits of the potential deal.

Who is winning this internecine public relations battle? The White House says recent polls show the president's message is resonating with Americans and that they are "optimistic" Obama will get the support he needs from Congress to sign the deal.

But Marsh, the strategist, thinks Warren is in control, shaping the debate and forcing the president to play defence. "You really have to make your case and make it stick, and right now in that category Elizabeth Warren is doing the better job," she said.