The pollster, Ben Tulchin, in a meeting with campaign aides, recommended a new offensive to influence older black voters, according to three people briefed on his presentation. The data showed two clear vulnerabilities for Mr. Biden: his past support for overhauling Social Security, and his authorship of a punitive criminal justice law in the 1990s.

But the suggestion met with resistance. Some senior advisers argued that it wasn’t worth diverting resources from Iowa and New Hampshire, people familiar with the campaign’s deliberations said. Others pressed Mr. Tulchin on what kind of message, exactly, would make voters rethink their support for the most loyal ally of the first black president.

Crucially, both Mr. Sanders and his wife, Jane, consistently expressed reservations about going negative on Mr. Biden, preferring to stick with the left-wing policy message they have been pressing for 40 years.