The possible increases have the support of the Army's senior leadership, which has been working to determine the mix of troops required to execute a strategy centered on a new combat formation.
The discussions at the Pentagon underscore the complex task the U.S. military faces as it prepares to deploy newly created combat advisory teams to some of the most violent, remote and heavily contested areas of Afghanistan.
The Obama administration, as part of its plan to wind down the Afghanistan war in 2015, limited advisers to higher headquarters far from the fighting. The new strategy that President Trump approved in August would push U.S. advisory teams to the battalion level, far closer to the front lines.
"This is a concept that got accelerated for Afghanistan, and it has been quite a process," a senior military official said of the plan to send the U.S. Army's first-ever Security Force Assistance Brigade to Afghanistan early this spring. "It has been a roller coaster." The official, who is involved in the troop planning, spoke on the condition of anonymity to discuss military planning.
Military officials said that some troops, particularly at the headquarters level, might come out of Afghanistan as new forces move into the theater and that they expect the total force this spring to be about 15,000 troops.
A spokesman for Gen. John W. Nicholson Jr., the top U.S. commander in Afghanistan, said U.S. Forces Afghanistan had not specifically asked for an increase in troop levels, suggesting the increase, if approved, would be considered an adjustment under the current plan rather than an increase associated with a shift in strategy.
The White House might want to weigh in on any plan to send additional troops to Afghanistan, according to U.S. officials.
Even as he signed off on the new war strategy, Trump has expressed misgivings about sending more resources into a conflict that has been grinding on with few signs of progress.
Senior administration officials said that the president has been known to affect an Indian accent and imitate Indian Prime Minister Narendra Modi, who in an Oval Office meeting last year told him, "Never has a country given so much away for so little in return" as the United States in Afghanistan.
To Trump, Modi's statement was proof that the rest of the world viewed the United States as being duped and taken advantage of in Afghanistan, these officials said.
Despite those misgivings, Trump has largely left execution of the war plan to the Pentagon, with little of the intense oversight that occurred during the Obama administration. At the time, Pentagon officials viewed the restrictions put on the campaign by the Obama White House as micromanagement.
But Trump has made it clear to senior Pentagon officials that he wants to see a quick return on the increased U.S. investment in troops and money in Afghanistan. Pentagon officials are also under pressure to keep troop numbers from growing significantly.
Nicholson said in the fall that the additional U.S. advisers and firepower will allow Afghan military and police forces to take control of 80 percent of the country in the next two years. Currently, the Afghan government controls about two-thirds of the country, with most of the rest controlled by the Taliban or contested.
The combination of the White House's insistence on quick progress and a desire to have the Afghan army take the lead in the fight has led to a sometimes heated debate inside the military over how best to support the Afghan forces.
At the core of the debate is the Army's new Security Force Assistance Brigade, a concept that was developed under the direction of Gen. Mark A. Milley, the Army's top general in the Pentagon. Unlike a traditional brigade, the unit does not come with any junior enlisted soldiers or junior officers and is broken up into 36 teams, each consisting of 12 soldiers, that can be parceled out among the forces they are advising.
The new unit is the product of lessons from the long insurgent conflicts following the Sept. 11, 2001, attacks, when the U.S. military has struggled to build cohesive and effective indigenous forces at a reasonable cost in places such as Iraq and Afghanistan.
"This is a test of the whole principle," retired Lt. Gen. David Barno, a former U.S. commander in Afghanistan, said of the brigade's upcoming deployment. The driving theory behind the new units is that specially trained advisers can mentor and assist foreign forces more effectively than regular combat forces.
Under the previous model, "it was always the first day of school" for regular soldiers showing up in Afghanistan to serve as advisers, Barno said.
The concept has drawn mixed reviews from the Army, which has long resisted the concept of advisory brigades that some Army officials worry are being built at the expense of more traditional brigades and will bleed combat power from the larger Army force.
Each 12-soldier team includes a medic, intelligence support and a person specially trained to call in airstrikes from circling U.S. Air Force planes.
But senior Army officials say that to be effective in Afghanistan, where the heaviest fighting occurs in remote areas, the teams need to bring extra support from attack helicopters, artillery units, intelligence troops and medical evacuation forces.
The additional forces would provide protection for U.S. troops that could be operating far from major cities and their higher headquarters. The extra capacity also increases their value to the Afghan forces.
"If you come with nothing and you don't provide extra firepower, aviation and [intelligence, surveillance and reconnaissance] support, then your value is questioned," the military official said. "It's one thing to provide advice, but firepower is something different."