The focus, Mr Emanuel says, should be on governance, not troops
The White House has said that it will take no decision on sending more troops to Afghanistan until it determines the new government is a "true partner".
White House Chief of Staff Rahm Emanuel told CNN TV it would be "reckless" to take such a decision without a thorough analysis of the new government.
Washington is debating a request for 40,000 more troops in Afghanistan.
Results from an investigation into fraud claims in the presidential poll are due in the next few days.
It is expected to reveal that incumbent President Hamid Karzai did not achieve more than the 50% of votes needed to avoid a second round.
AFGHAN FRAUD ALLEGATIONS
13 Oct: Karzai casts doubt on fair functioning of ECC, but his opponents accuse him of manufacturing his concerns
30 Sep: UN recalls envoy Peter Galbraith following row over the vote recount process
15 Sep: ECC chief says 10% of votes need to be recounted
8 Sep: IEC says votes from 600 polling stations "quarantined"
3 Sep: Claims 30,000 fraudulent votes cast for Karzai in Kandahar
30 Aug: 2,000 fraud allegations are probed; 600 deemed serious
20 Aug: Election day and claims 80,000 ballots were filled out fraudulently for Karzai in Ghazni
18 Aug: Ballot cards sold openly and voter bribes offered
Mr Karzai insists he won the 20 August vote but EU observers say that as many as one in four votes cast was suspicious.
Foreign officials - including US Secretary of State Hillary Clinton and France's Foreign Minister Bernard Kouchner - have been pressing Mr Karzai to accept that he might have to face a run-off.
Speaking to CNN, Mr Emanuel said the US would want first to be sure that the government was capable of becoming a "true partner" able to govern the country.
"It would be reckless to make a decision on US troop levels if, in fact, you haven't done a thorough analysis of whether, in fact, there's an Afghan partner ready to fill that space that the US troops would create and become a true partner in governing the Afghan country," he said.
His comments were echoed by Senator John Kerry, who said that "it would be entirely irresponsible for the president of the United States to commit more troops to this country, when we don't even have an election finished".
Gen Stanley McChrystal, the US and Nato commander in the country, has recommended sending the extra troops as the US reviews its strategy.
Hamid Karzai is said to be angry at the prospect of facing a run-off
But Mr Emanuel said that the number of soldiers deployed in the country was secondary to whether or not they could work alongside an effective Afghan administration.
"The question does not come how many troops you send, but do you have a credible Afghan partner for this process that can provide the security and the type of services that the Afghan people need?" he told CNN's State of the Union programme.
Initial results from August's election gave Mr Karzai 55% of the votes, with his nearest rival, Abdullah Abdullah, getting 28%.
Jonathan Marcus, diplomatic correspondent
This explains what some have seen as President Barack Obama's procrastination and delay - it is intended to make President Karzai sweat.
The problem is, who really has who over a barrel? Washington can up the pressure but if Mr Karzai fails to deliver, what then?
Walking away from Afghanistan is not one of the options under consideration - though a significant scaling back of US ambitions and resources could eventually be on the cards if no legitimate governing formula can be found.
But the UN-backed Electoral Complaints Commission (ECC) launched an investigation into the vote following allegations of widespread electoral fraud.
It will report to the Independent Election Commission (IEC), which could adjust the final tally, bringing Mr Karzai's vote total below 50% and triggering a run-off.
Officials say Mr Karzai is furious over the prospect of facing a second round, threatening to delay or block attempts to hold a second round.
He has refused to accept the ECC's findings before they are released.
The ECC had been expected to announce its findings on Saturday. But the reported confrontation with Mr Karzai may delay the official announcement of results.