An announcement could come as early as Saturday but more likely Sunday, the person said.
Abdullah, who was once Karzai's foreign minister, put forward several condition earlier this week for the Nov. 7 election to be credible, but intensive talks between the two candidates and their supporters over the past few days broke down Friday, the person, a Westerner, said on condition of anonymity because the discussions were private.
The political stalemate in Kabul comes as President Barack Obama has been meeting with his advisers to try to determine U.S. strategy in Afghanistan, including troop levels. A weakened Afghan government will make it harder for Obama to get public support for his efforts.
The runoff election in Afghanistan became necessary after widespread fraud in the first-round of voting in August resulted in thousands of Karzai's ballots being invalidated, pushing him below the required 50 percent margin to win. Concerns have been raised about a possible repetition of the ballot-box stuffing and distorted tallies in the second round.
Abdullah complained Monday that there were no assurances that the November vote would be fairer than the first balloting.
One of Abdullah's demands was to fire the head of the Karzai-appointed Independent Election Commission, Azizullah Lodin, within five days.
He said Monday that Lodin has "no credibility." But Lodin has denied allegations of bias in favor of Karzai, and the election commission's spokesman has already said Lodin cannot be replaced by either side.
Abdullah also wanted changes in several ministries and a power-sharing deal, the person close to the talks told The Associated Press.
Abdullah was pressing for a power-sharing agreement with Karzai instead of a vote, but Karzai refused, insisting instead on a vote and then a power-sharing agreement, the person said.
Despite the massive fraud and rejected ballots, Karzai's vote in the first round was far higher than Abdullah's and he is widely expected to win the runoff.
This year's election - the first run by Afghans since the ouster of the Taliban - was supposed to affirm the government's credibility. Instead, the massive fraud has undermined the legitimacy of the Karzai administration just as U.S. officials are debating whether to send more troops.
The Taliban, who threatened voters during the August balloting, have warned Afghans that they risk further attacks if they do not stay away from the polls next week. On Wednesday they targeted a U.N. guest house where 34 staff - including a number of U.N. election workers - were sleeping. Eight people were killed in the assault, five of them U.N. staff members.