Warren tied Biden with white college graduates at 23%, and was also essentially tied with him among the most liberal quarter of Democratic voters.
But her support fell to 9% among white nongraduates of college, 6% among nongraduates of all races, 3% among all African Americans and 7% among all Latinos, the poll found.
Warren "can't have 6% of people without college degrees for her" and win, said Bob Shrum, director of USC's Center for the Political Future, which co-sponsored the poll. "She needs to build that up."
The findings also offer some backing to Biden's argument that he would give the Democrats their strongest chance in an election against President Donald Trump: Potential swing voters in the general election see Biden as relatively close to their own position but view the other leading Democrats as further to the left, the poll finds.
The poll also finds:
Health care policy, which has been a big focus for arguments at the Democratic debates, has not become a litmus test for voters.
The poll asked potential Democratic primary voters if they support a "Medicare for All" plan that would do away with private insurance, as backed by Sanders and Warren, or a plan that would allow people to either buy into Medicare or keep their existing insurance, as Biden advocates. The largest group, more than 4 in 10, said they support both. An additional 19% support only the optional plan, while 14% support only the full public plan and 19% haven't heard enough to say.
Among potential general-election voters, the results are less equivocal: The Medicare option draws wide support; the full public plan generates greater opposition. By 48% to 14%, with 38% uncertain, potential general election voters support the option; the full public plan draws 39% in favor, 34% opposed and 26% uncertain.
Support and opposition to Medicare for All divides deeply along partisan lines while the Medicare option does not, noted Jill Darling, the survey's director. But if the Medicare option becomes a general election issue, views could polarize, and many of those currently on the fence could shift against it, she said.
For now, however, the poll suggests that advocating an end to private insurance "could easily be a real burden for Democrats in the general election," said Shrum.