NATO officials said the request by General David Petraeus to bolster the International Security Assistance Force (ISAF) was made last week and included a call for about 750 more personnel to train the Afghan security forces.
"It has been determined that around 2,000 forces will be required," a NATO official said, speaking on condition of anonymity. "There is an ongoing discussion on this issue."
The request comes before a visit to Washington on Monday and Tuesday by NATO Secretary-General Anders Fogh Rasmussen.
Officials at Petraeus' NATO command in Kabul declined to comment and referred questions to NATO headquarters in Brussels.
One NATO official said it was assumed the United States would offer more troops to the 150,000-strong foreign force in Afghanistan, but the alliance was looking for contributions from non-US members and partner countries too.
NATO has been seeking to bolster its effort to train the Afghan armed forces, but has struggled to persuade its 28 members and allies contributing to ISAF to commit the necessary personnel as public support for the war wanes.
NATO's Afghan mission was expected to top the agenda in Rasmussen's talks with US President Barack Obama, National Security Adviser James Jones and other administration officials.
Obama has sent tens of thousands more troops to Afghanistan to create conditions to expand Afghan forces, but has said the extra troops could be gradually withdrawn from July 2011.
Critics say this strategy has backfired, sending a signal to the Taleban that the United States is preparing to wind down at a time when NATO forces are suffering record casualties.
The number of foreign troops killed this year has reached at least 500, compared with 521 in all of 2009, according to an independent monitoring site on Monday and a Reuters tally.
Petraeus said last week the withdrawal of US troops from next July would begin with a general "thinning out" of forces rather than any large-scale drawdown.
The stepped-up NATO training effort envisages boosting the size of the Afghan army to 171,600 by October 2011 and the police force to 134,000.
Afghan President Hamid Karzai has set a goal of 2014 for Afghan forces to take over full responsibility for security from foreign troops, but the deadline will rely heavily on the success of foreign troops in battling Taleban insurgents and training the Afghan army.
Rasmussen hopes at a summit in November to set a target date in 2011 for starting switching responsibility for security to Afghan forces. Until midway through this year he had said he hoped the process could start this year.
