Multiple times each day, President Biden dials up Mike Donilon, a close adviser since the 1980s, to chew on the latest polls and headlines.

“What’s your instinct? What do you think?” Mr. Biden will ask Mr. Donilon, who recently left the White House for the campaign’s Delaware headquarters.

Once a week, Mr. Biden summons Ron Klain, his former chief of staff, to workshop the best attacks to use against former President Donald J. Trump as the presidential debate draws closer.

When he leaves for Delaware on weekends, Mr. Biden seeks out Ted Kaufman, a confidant who represents the president’s ties to the state that introduced him to the national stage more than a half-century ago. It was Mr. Kaufman who was brutally direct with Mr. Biden when a plagiarism scandal threatened his first campaign for president in 1987.

“There’s only one way to stop the sharks,” Mr. Kaufman told him at the time, “and that’s pull out.” Mr. Biden did.

Interviews with dozens of people close to the president reveal a truth at the heart of Mr. Biden’s political life: While he is surrounded by a diverse and multigenerational crowd of campaign operatives, policy experts and cabinet secretaries, he reserves his full trust for a small circle of insiders who are the definition of old school.