Administration officials in recent days have said they are considering a range of options for a residual U.S. troop presence of as few as 3,000 and as many as 15,000, with the number linked to a specific set of military-related missions like hunting down terrorists.
Asked in a conference call with reporters whether zero was now an option, Rhodes said, "That would be an option we would consider."
His statement could be interpreted as part of an administration negotiating strategy. On Friday Karzai is scheduled to meet Obama at the White House to discuss ways of framing an enduring partnership beyond 2014.
The two are at odds on numerous issues, including a U.S. demand that any American troops who would remain in Afghanistan after the combat mission ends be granted immunity from prosecution under Afghan law. Karzai has resisted, while emphasizing his need for large-scale U.S. support to maintain an effective security force after 2014.
In announcing last month in Kabul that he had accepted Obama's invitation to visit this week, Karzai made plain his objectives.
"Give us a good army, a good air force and a capability to project Afghan interests in the region," Karzai said, and he would gladly reciprocate by easing the path to legal immunity for U.S. troops.
Karzai is scheduled to meet Thursday with Panetta at the Pentagon and with Secretary of State Hillary Rodham Clinton at the State Department.
Without explicitly mentioning immunity for U.S. troops, Obama's top White House military adviser on Afghanistan, Doug Lute, told reporters Tuesday that the Afghans will have to give the U.S. certain "authorities" if it wants U.S. troops to remain.
"As we know from our Iraq experience, if there are no authorities granted by the sovereign state, then there's not room for a follow-on U.S. military mission," Lute said. He was referring to 2011 negotiations with Iraq that ended with no agreement to grant legal immunity to U.S. troops who would have stayed to help train Iraqi forces. As a result, no U.S. troops remain in Iraq.