The proposals are the result of a comprehensive administration review of the U.S. strategy for Afghanistan, where the United States now has 8,400 troops.
There are an additional 5,000 forces from NATO countries assisting with the mission.
Asked at an Senate Armed Services Committee hearing in February to describe the fight against Taliban forces, Army Gen. John Nicholson responded, "I believe we're in a stalemate."
He added that he needed more troops from the United States or NATO countries to train, advise and assist the mission, saying, "We have a shortfall of a few thousand."
The proposals could lead to the addition of 1,000 to 3,000 U.S. troops to the existing forces, according to the U.S. official. NATO partners will be requested to provide a similar number of personnel. The number of additional U.S. personnel will likely depend on how many troops are committed by NATO countries.
Additional troop levels will likely be a topic of discussion at the May 25 NATO summit in Brussels.
White House press secretary Sean Spicer said Tuesday that no decision had been made on Afghanistan and that Trump was reviewing the U.S. strategy there.
"He's trying to figure out, walk back from a goal of eliminating this threat and then tell me how we get there, as opposed to tell me how many troops we need and what you're going to do with them," Spicer said.
"The president is asking to relook at the entire strategy and then figure out what the footprint is, in a variety of ways to get there," said Spicer. "That is a different look at what the strategy is, versus what it had been."
Other proposals being presented to the White House call for the Pentagon to set the U.S. troops levels and expand the permissible conditions for airstrikes against the Taliban, according to another U.S. official.
At the time, a U.S. defense official said the same authority had not been expanded to Afghanistan, pending the administration's review of its overall strategy in that country.
Then-President Obama last summer approved broader authority for U.S. forces, allowing close air support for Afghan troops' offensive operations and permitting conventional forces to accompany Afghan troops on certain occasions.
Such changes loosened restrictions Obama put in place in late 2014 as the NATO combat mission in Afghanistan transitioned to a training mission. Among other things, the restrictions limited airstrikes against the Taliban to extreme self-defense situations and barred targeting Taliban fighters in airstrikes simply because they are members of the group.
As Afghan forces began bearing the brunt of the fighting, they suffered significant casualty rates. Over the past two years, Afghan military and police have averaged more than 5,000 personnel killed each year in fighting with the Taliban.
Trump has had little to say about Afghanistan since he assumed office. As a candidate, he questioned the role of U.S. troops in Afghanistan.
The U.S. military presence in Afghanistan began Oct. 7, 2001, in response to the 9/11 terrorist attacks.
The number of U.S. troops in Afghanistan remained low compared with the major operations in Iraq, which began in 2003. But after a surge of forces in 2009, it reached a peak of 100,000 in 2010 under Obama.
His administration set in place timelines for a reduction in U.S. troop levels as Afghan forces became responsible for their own security.