Most of the president’s advisers see Mr. Sanders, if he were to become the nominee, as helping to solve Mr. Trump’s problem with suburban voters in states like Virginia, where the 2018 midterm elections showed that moderate and independent voters have recoiled from the president’s behavior, controversies or policy positions.

But some have concerns that Mr. Sanders might be more durable in the Rust Belt states, like Michigan and Wisconsin, with high concentrations of white working-class voters that emerged as trouble spots for the president in the 2018 elections.

The advisers say that in their voter research Mr. Sanders registers with his own supporters as authentic — the same quality that Mr. Trump’s base ascribed to the president in 2016. They view Mr. Sanders as a more difficult opponent than Senator Elizabeth Warren, Democrat of Massachusetts, but less of a challenge than Mr. Biden.