Military leaders would also regain the authority to target Taliban leaders with air strikes under the proposals.
President Donald Trump has not approved the plan, unnamed officials say. They may include a request that other Nato countries send 3,000-5,000 soldiers.
There are 13,000 Nato troops currently in the country, 8,400 of them US.
US combat operations against the Taliban officially ended in 2014, but special forces have continued to provide support to Afghan troops.
In February, the commander of US troops in Afghanistan Gen John Nicholson told a Senate committee there was "a shortfall of a few thousand".
He said he needed more troops to break a "stalemate".
Last month, the Taliban announced the start of their "spring offensive" a week after killing at least 135 Afghan soldiers in a military compound near the northern city of Mazar-e Sharif.
The group said it would use military alongside political tactics and that its main target would be foreign forces.
Taliban militants this week seized a district in northern Afghanistan in their continuing attempts to take the whole city of Kunduz. Thousands of families have been forced to leave their homes.
Has learnt that three key Taliban commanders, including the head of a new commando unit, have been moved from Helmand - which is already largely Taliban-controlled - to Kunduz for the offensive.
The previous US President, Barack Obama, set deadlines for cutting the numbers of US soldiers in the country and removed the Pentagon's authority to directly target Taliban leadership.
However, he was forced to abandon the troop reduction target.