These Democrats worry that her uncompromising liberalism would alienate moderates in battleground states who are otherwise willing to oppose the president. Many fear Ms. Warren’s past claims of Native American ancestry would allow Mr. Trump to drown out her policy message with his attacks and slurs against her. They cite her professorial style and Harvard background to argue that she might struggle to connect with voters from more modest circumstances than hers, even though she grew up in a financially strained home in Oklahoma.

And there are Democrats who, chastened by Hillary Clinton’s defeat in 2016, believe that a woman cannot win in 2020.

“I think she’s terrific but my questions about her are, can she get elected with the negativity, with all the stuff that’s thrown at her?” asked Rick Morris, a New Hampshire carpenter who attended a house party for Ms. Warren there last month. “Usually in the primary I vote for whoever I like the most, but this one I will put in electability.”