Both Carol Moseley Braun and Congressman Danny Davis rejected a plea from Rev. Jesse Jackson, the president of Rainbow PUSH, for one of them to drop out.
Jackson summoned Davis, Braun and others for a late night meeting at the Rainbow PUSH offices Wednesday.
While they spoke about issues concerning poor Chicagoans, Jackson tried persuading the two candidates to get behind one African American candidate.
"The fact is two can lose; only one can win, and they must make the determination ultimately as for the greater good," said Jackson.
The idea of a consensus candidate goes back to 1983, when Harold Washington was elected.
Braun and Davis have so far declined Jackson's suggestion.
"I'm not going anywhere," said Braun. "I'm going to keep campaigning."
"The good part of democracy is about the business of choices," said Davis.
Jackson says a split vote is inevitable because Braun and Davis represent similar interests and values and appeal to the same audience.
"You are less able to motivate people, you're less able to raise funds, less able to build infrastructure," said Jackson.
Braun, however, says her appeal is to all Chicagoans. The former U.S. Senator says she has the record to prove it.
"I'm very happy about the kind of response I've gotten across the city in previous elections," said Braun.
Davis says he is open to continuing the conversation about a possible consensus candidate, but that it is more about losing support.
"I think when people reach the point when they figure they cannot win, that's kind of when they decide to do something else, and I don't' think anyone has decided yet that they can't win," said Davis.
The idea of a consensus candidate for any race is something mayoral hopeful Gery Chico says he does support.
"I have said from the beginning - I haven't mattered much about who is in the race, who's out of the race, I haven't tried to talk her in, talk her out - I don't subscribe to that," said Chico.
Chico's views are similar to those of Miguel del Valle, the other major Latino candidate in the race.
Del Valle says any candidate who wants to be mayor must also be able to transcend the traditional demographic lines.
Meanwhile, Jackson says the conversation with Davis and Moseley Braun will continue.