Mrs Clinton told reporters in Washington that Mr Karzai was "going to announce his intentions" on Tuesday, in response to the investigation by the UN-backed Electoral Complaints Commission (ECC).
Mrs Clinton did not indicate what Mr Karzai's response might be, but said she was "encouraged at the direction that the situation is moving".
Meanwhile Mr Karzai's own spokesman, Waheed Omar, said the president would wait for the results to be announced by the Afghan-led Independent Electoral Commission (IEC).
The IEC is widely regarded as pro-Karzai, and it is not clear whether it will accept the ECC's findings.
In its much-anticipated report on Monday, the ECC said it had found "clear and convincing evidence of fraud" and deducted hundreds of thousands of votes from the main candidates.
Hillary Clinton: "I am very hopeful that we will see a resolution"
Its investigation focused on 600 of the most serious complaints, and a sample audit of suspect votes at 3,377 polling stations. At 210 polling stations all the ballots were invalidated.
Under the Afghan constitution, the EEC's report passes to the Afghan-led Independent Election Commission (IEC) - appointed by Mr Karzai - which can decide whether to call for a run-off.
The Afghan president has previously refused a run-off, insisting he won the election outright.
Initial results released last month gave him nearly 55% of votes, and Mr Abdullah 28%.
Lyse Doucet, BBC News, Kabul
When this tangled process began, the ECC, the only electoral body composed of Afghan and foreign representation, was regarded as the "final arbiter".
The IEC's role was to ratify and announce the results. The IEC is accused by many of being too close to a president who appointed all its commissioners. It now says it needs "a day or two" to examine the details of the ECC report.
One source warned of a possible "train crash". Others are still hoping a compromise can emerge at this critical 11th hour.
President Karzai has repeatedly warned foreign countries not to interfere in the election process.
Sources say he firmly believes Western countries, in particular the United States and Britain, are conspiring to rob him of victory.
The ECC launched its investigation in the wake of the 20 August vote as allegations of mass fraud began to emerge.
The panel reports to the IEC, which will make the final announcement on the election outcome.
Diplomats have accused the IEC of stalling to give the president more time to reach a deal with Mr Abdullah, possibly on power-sharing to avoid a run-off.
The BBC's Barbara Plett at the United Nations says the Americans - and their Nato allies - are looking for a second round or a national unity government to inject some legitimacy into the Afghan elections.
They say the government must be legitimate if the US public can be convinced it is worth sending more soldiers to fight in Afghanistan.
With violence at its worst levels across Afghanistan since the Taliban were ousted in 2001, there are warnings the ongoing political paralysis will only embolden the militants.
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