The troop levels under consideration remain tentative but the official, who spoke on condition of anonymity, said options under consideration range from 6,000 to 15,000 American boots on the ground.
"It's in that ballpark," the official said, confirming an earlier report by the Wall Street Journal.
A Pentagon spokesman declined to discuss possible troop numbers after the NATO-led combat mission formally ends by 2015. But Defense Secretary Leon Panetta said earlier this month that the White House was studying options from the US commander in Afghanistan, General John Allen, and would be making a decision within a "few weeks."
The follow-on force would carry out counter-terrorism operations against Al-Qaeda, and provide training and logistical support for Afghan forces, Panetta said on November 12, speaking to reporters during a trip to Asia.
Although Washington has committed to pulling out the bulk of the 67,000 US troops now in Afghanistan by the end of 2014, the United States also has promised to keep a follow-on force on the ground under an agreement with Kabul.
But Obama has yet to announce how many troops would stay in Afghanistan beyond 2014, and retaining a force in the country would require delicate negotiations with the Afghan government to resolve legal issues and access to bases in the country.
"Any US presence will only be at the invitation of the Afghan government and aimed at training Afghan forces and targeting the remnants of Al Qaeda," Little said.
Once Obama decides how many troops to keep in the country after 2014, Allen is due to issue recommendations soon on the pace of the planned troop drawdown in the run-up to the 2014 deadline.
Apart from American troops in Afghanistan, there are about 37,000 forces from NATO and coalition partners in the country, while the Afghan government has 337,000 soldiers and police.
US and Afghan officials launched talks on November 15 to hammer out a status of forces agreement that would allow American forces to stay on after 2014.
US officials and military officers have stressed that Washington does not seek any permanent bases in Afghanistan but they will be keen to secure access to airfields and bases while extracting promises of legal protection for American soldiers stationed in the country.
However, Afghan President Hamid Karzai has warned of possible disagreement over the sensitive issue of immunity from prosecution for any American or NATO soldiers deployed after 2014.
In Iraq, the United States pulled out all its troops, leaving no residual force, after failing to persuade Baghdad to grant its soldiers immunity from prosecution in local courts.