Even former Vice President Joseph R. Biden Jr. does not expect to secure Mr. Obama’s backing if he runs, according to allies of Mr. Biden’s…

During these informal conversations, usually at his office in Washington, Mr. Obama has offered a combination of supportive advice and sober warnings, cautioning candidates that running for president is a more punishing process than they could ever imagine, according to seven people who have spoken with him directly or were briefed in detail on the meetings.

Mr. Obama continues to express frustration that he did not anticipate Mr. Trump’s victory, these people said, even after years of clashing with the forces of right-wing populism as president. He has urged candidates to push back on Mr. Trump’s bleak and divisive rhetoric about economic change, and to deliver a competing message that can resonate even in Republican-leaning areas, courting rural voters and other communities that tend to distrust Democrats.