The United States currently has 37,500 troops in the country. A security agreement was negotiated last year with President Hamid Karzai that would allow some to remain, but Karzai has not yet signed it and is demanding changes.
The Times said officials in the Obama administration have debated a force ranging from 8,000 to 12,000 with the Pentagon settling on the midpoint.
"The proposal is 10,000 or basically nothing, a pullout," one unnamed official told the newspaper.
While the U.S. State Department supports the Defense Department, Vice President Joe Biden and some others in the administration have argued a smaller force could do the job, the Times said.