“It would energize the progressive side of the party that many of us are part of,” the Rev. Al Sharpton said of a possible win by Mr. Garcia. “The apprehension is that the right-wing media would take it as some people abandoning the president’s endorsement. That’s why it gives some of us pause. We don’t want to feed into some of the hysteria against the president.”

Mr. Sharpton, the longtime civil rights activist, said that as he considered whom to endorse, he was faced with “a dicey choice.”

“I’m attracted to Chuy’s populism and his consistent progressive base. He goes all the way back to Harold Washington,” Mr. Sharpton said, referring to a 1980s mayor of Chicago. “But Rahm is good with the black business community, and he’s been good on some of the stuff when he was with the president as chief of staff.”

Mr. Emanuel is counting on a strong showing from African-Americans, who make up 33 percent of the city’s population. Some of Mr. Garcia’s worst results in the initial balloting came from heavily black wards.

Another challenge for hard-line progressives is that Mr. Emanuel, while heavily funded by business interests and some wealthy Republicans, has a record of accomplishment that covers many liberal priorities. He may have angered the teachers union and many black residents by closing 50 failing schools in mostly minority neighborhoods. But he passed full-day kindergarten, a move similar to Mr. de Blasio’s expansion of prekindergarten. Mr. Emanuel also added free community college for high-achieving public school graduates. He devoted funds to improving service on the El train in some South Side neighborhoods. He helped raise the minimum wage, a big issue for the left.

That such achievements have not been enough to mollify his critics illustrates the expectations of a resurgent political left. His critics in the Democratic Party say that he agreed to raise the minimum wage only under pressure and that Mr. Garcia had favored a higher increase.