An easy path to the nomination could allow Mrs. Clinton to enter a general election with more funding than the Republican nominee, who would have had to spend heavily to beat a wide field of competitors. Ms. Warren represents Republicans’ best hope for an expensive, prolonged battle for the Democratic nomination, weakening Mrs. Clinton along the way, political operatives on both sides say.

That desire appears to trump the fact that Ms. Warren’s views about taxation, regulation and the role of government are so at odds with Republican tenets. “There is nobody in this country who got rich on his own,” she famously said in 2011.

Ms. Warren told Fortune magazine this month that she would not run to succeed President Obama, but that has not stopped speculation.

“Elizabeth Warren says, ‘I’m not running; I don’t want to be president,’ ” the radio host Rush Limbaugh said recently. “Translation: ‘I can’t wait and I am running. But I’m just not going to admit it right now.’ ”

Republicans said Ms. Warren would deliver a perfect “trifecta” in diminishing Mrs. Clinton. She attracts young, liberal supporters who view Mrs. Clinton as too centrist. A Warren candidacy would take away a central theme expected of Mrs. Clinton’s campaign — that it is time to elect a female president. And Ms. Warren’s presence in the primary season could push Mrs. Clinton to adopt liberal positions that might turn off independents in a general election.

It first became apparent that Ms. Warren could be an effective tool in moving Mrs. Clinton off message when the two appeared at a joint rally in October for Martha Coakley, the Democratic nominee for governor of Massachusetts.

Image As a former secretary of state and 2008 presidential candidate, Hillary Rodham Clinton could emerge from the primary season relatively unscathed. Credit... Liam Richards/The Canadian Press, via Associated Press

In her speech, Mrs. Clinton tried to channel some of Ms. Warren’s populist zeal but flubbed a variation of the senator’s controversial line about the roots of success. “Don’t let anybody tell you that, you know, it’s corporations and businesses that create jobs,” Mrs. Clinton said.