Some of Mr. Trump’s advisers believe that there isn’t a downside in attacking Mr. Obama, who is vastly more popular than Mr. Biden. They argue that Mr. Trump challenged Mr. Obama’s record throughout the 2016 campaign, and they take Mr. Trump’s victory as affirmation of his approach.

But the president is now attacking Mr. Obama’s integrity, not just his policies, a move that makes some of Mr. Trump’s advisers anxious. In a survey of voters in 17 battleground states commissioned by the Republican National Committee in March, Mr. Obama was deeply popular, especially compared with current elected officials. Sixty percent of respondents said they viewed Mr. Obama favorably, compared with 36 percent who said they saw him negatively, according to a person briefed on the data.

Meanwhile, Mr. Trump’s advisers are cognizant that the virus is harming some of the black voters to whom the president had hoped to sell himself as the person who would help build a better economic life.