Afghan lawmakers have been vocal about the reduced numbers as well.
"If Americans pull out all of their troops without a plan, the civil war of the 1990s would repeat itself," Naeem Lalai, an outspoken lawmaker from volatile Kandahar province told Reuters on Wednesday.
Mark Jacobson, a former NATO adviser to then-U.S. commander in Afghanistan, Gen, David Petraeus, said numbers under 6,000 troops would make it impossible to conduct any kind of mission in the country.
But Jacobson says the numbers being floated are also a shot across Karzai's bow.
"Karzai's people have said the U.S. needs Afghanistan more than Afghanistan needs U.S. troops and the administration is less than pleased with that," Jacobson said.
U.S. defense officials are playing a wait-and-see game on what the Karzai visit will produce.
A Republican member of the Senate Armed Service Committee, Lindsay Graham of South Carolina, said on Sunday that he would like to see a much larger enduring U.S. troop presence in Afghanistan.
Graham's comments represent what the opposite side of the spectrum, generally embraced by Republicans.
"I think somewhere in the 15,000 to 20,000 range, depends on what the military commanders say," Graham told CNN's Candy Crowley on State of the Union
But Jacobson says we the numbers could also depend on how the Afghan government is perceived by the Afghan people. If the government seems to be providing basic services and has the support of the people, a reduced U.S. troop number could be appropriate.
Combined with a competent Afghan security force, there could be little reason to keep troops other than a U.S. counterterrorism force inside the country, according to Jacobson.