The Democrats are betting that if anyone can pull it off, it will be the man who burst into the consciousness of many Americans at the 2004 convention in Boston. It was then, as a state senator in Illinois who was running for the U.S. Senate, that he was selected to be the keynote speaker. He labored over the speech, he wrote later, drafting it longhand. The resulting speech left a bigger impression on his audience than the subsequent acceptance speech of the party’s nominee, Senator John F. Kerry, who, like Mrs. Clinton, later served as Mr. Obama’s secretary of state.

Mr. Obama’s line that evening, that “there is not a liberal America and a conservative America — there is the United States of America,” promised a vision of national unity at a time that the seams were just being stretched. “There is not a Black America and a white America and Latino America and Asian America,” he added. “There’s the United States of America.”

Former aides to Mr. Obama say he is bound to return to that theme on Tuesday night, as he makes the case for not fueling the societal divisions Mr. Trump has at once benefited from and fed. And his former aides expect him to draw heavily on his old critique of Mr. Trump — “does anyone really believe that a guy who’s spent his 70 years on this Earth showing no regard for working people is suddenly going to be your champion?” — updated with a few indictments and a conviction.

“Donald Trump calls our military a disaster,” Mr. Obama said in 2016, pressing his case. “Apparently, he doesn’t know the men and women who make up the strongest fighting force the world has ever known. He suggests America is weak. He must not hear the billions of men and women and children, from the Baltics to Burma, who still look to America to be the light of freedom and dignity and human rights.”

And, Mr. Obama said, Mr. Trump “cozies up” to President Vladimir V. Putin of Russia and “tells our NATO allies that stood by our side after 9/11 that they have to pay up if they want our protection.”

Mr. Axelrod noted that “so much has happened since 2016.”

“And what was then just speculation about Trump’s potential for excesses and trespasses,” he said, “are now a part of his history as president an aggrieved former president.”