Speaking to reporters traveling with him to Munich, where he will attend a security conference, Pompeo expressed both optimism and caution.
The tentative timeline came a day after Afghan President Ashraf Ghani said there had been a possible breakthrough in US-Taliban talks in Qatar.
A US-Taliban peace deal, which sources said could be signed this month, would pave the way for a withdrawal from Afghanistan of some 13,000 US troops and thousands of other North Atlantic Treaty Organisation personnel, 18 years after a US-led coalition invaded following the September 11, 2001, Al Qaeda attacks on the United States. Trump has said it's time for American troops to come home, but a withdrawal of North Atlantic Treaty Organisation or USA forces could result in further instability and violence in the troubled country.
One of them: USA negotiators will likely book the best to come to the support of all Afghan forces with air power.
The official said the agreement for a seven-day "reduction in violence" is "very specific" and covers the entire country, including Afghan government forces.
"If it holds - and that is a big if - then 10 days later there would be negotiations between the Afghan government and the Taliban about the way ahead in Afghanistan", Fox News' Jennifer Griffin said on "America's Newsroom".
Should the Taliban comply, the "reduction in violence" agreement would be followed by the signing of an agreement that would initiate peace negotiations that include all Afghan sides.
"I think we're very close", he told Geraldo Rivera's Roadkill podcast on Thursday (local time)". The Taliban, "they'd like to make a deal, too".
President Trump had indicated that the USA was close to working out a deal.
"We are taking a hard look at it".
"I think it also bears underscoring that [this process] upholds the human rights of all citizens, including the gains made in regards to women's rights and freedoms and leads to sustainable peace in Afghanistan".
"Should the Talibs implement what they've committed to doing, we'll go forward with the agreement", the official said.
"We're obviously following very closely what has been going on in these talks and our office in Kabul's been kept up to date", United Nations spokesman Stephan Dujarric said in answer to a question from Anadolu at a press briefing on Friday.
The final details were hammered out in recent days by U.S. special envoy for Afghanistan Zalmay Khalilzad and Taliban representatives in Doha, Qatar. 'I am not looking to kill hundreds of thousands of people in Afghanistan, many of them totally innocent'.
Between the signing of the U.S. -Taliban deal and starting of intra-Afghan negotiations, the insurgent group and Afghan authorities would be expected to release each other's prisoners. Taliban insurgents have a strong presence in the region.