The prospective deal would see the USA pull thousands of troops from Afghanistan, while the Taliban would provide security guarantees and launch eventual talks with the Western-backed government in Kabul.
President Donald Trump is gambling that direct engagement with the Taliban, which once provided a safe haven in Afghanistan for the al-Qaeda terrorist group, will help him meet a pledge when running for office in 2016 to get America out of what he called "endless wars". In September, Trump abruptly called off the talks in response to a suicide bombing in Kabul that killed an American soldier.
Mr Pompeo on Thursday hailed recent progress, but said the negotiations were complicated and that a peace deal had not yet been reached.
Speaking at the same North Atlantic Treaty Organisation 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".
"We've said all along that the best, if not only, solution in Afghanistan is a political agreement", Esper told reporters in Brussels.
"It will be a continual evaluative process as we go forward - if we go forward", said Esper. "Progress has been made on that front and we'll have more to report on that soon, I hope", Esper said.
Mr Trump has made withdrawing United States troops from Afghanistan a key foreign policy aim.
There are some 12,500 USA troops in Afghanistan, as well as thousands of European forces participating in the NATO-led Resolute Support mission.
In December 2018, the Taliban announced they would meet U.S. officials to try to find a "roadmap to peace".
Finally, the U.S. -Taliban talks themselves have been hard, but they are nothing compared to the complex peace, reconciliation and power-sharing negotiations to come between the Taliban, the Afghan government, and other Afghan political stakeholders.
"We're at a pivotal point in this process, because the two sides are now closer than they've ever been before to a deal".
But days later, Mr Trump said the talks were "dead", after the militant group admitted to killing a U.S. soldier. "They have to make real compromises around the negotiating table".
In the months since the deal collapsed, there has so far been no let-up in fighting.
The announcement came a day after Afghan President Ashraf Ghani reported "notable progress" in negotiations with the Islamist insurgents.
"'This year like other years we don't have any concerns about any location that will be under Taliban threats -we will follow our priorities", Khalid added.