Esper said on February 13 a drawdown of troops and further negotiations with the militants would be "conditions-based" and begin after a decline in violence.
"The United States and the Taliban have negotiated a proposal for a seven-day reduction in violence," Esper said.
"We've said all along that the best...solution in Afghanistan is a political agreement," he added. "It will be a continual evaluative process as we go forward -- if we go forward."
Progress toward reducing violence could usher in direct peace talks between the militants and the Afghan government to end the nearly two-decade war.
NATO Secretary-General Jens Stoltenberg the day before reiterated that the alliance "fully supports the U.S.-led peace efforts, which can pave the way for intra-Afghan talks."
Speaking at the same NATO summit in Brussels attended by Esper, the secretary-general said the militants "need to demonstrate that they are both willing and capable to deliver a reduction of violence and contribute to peace in good faith."
He said the militant fighters have to "understand that they will never win on the battlefield. They have to make real compromises around the negotiating table."
Earlier, Afghan President Ashraf Ghani tweeted he had been informed by U.S. Secretary of State Mike Pompeo of "notable progress" in talks between U.S. and Taliban negotiators, which have now lasted for more than a year.
The prospective deal would see the U.S. pull thousands of troops from Afghanistan, while the Taliban would provide security guarantees and launch eventual talks with the Western-backed government in Kabul.
There are some 12,500 U.S. troops in Afghanistan, as well as thousands of European forces participating in the NATO-led Resolute Support mission.