He said their meeting was "productive".
Meanwhile, according to the same Egyptian source, some Arab capitals are set to share their concerns with Pompeo over Trump's decision to withdraw United States troops from Syria that seemed to be giving a green light to Ankara.
The official, speaking on condition of anonymity, said releasing the fighters, among them Europeans and some USA citizens, would be "unacceptable" since they could simply rejoin the remnants of IS fighters in Syria or elsewhere. Halbusi discussed US efforts to help Iraq "with reconstruction and investment".
The terms included total defeat of IS - still active in some Syrian regions - and ensuring that Kurdish fighters who fought alongside the Americans against the jihadists will be protected.
Air Force Col. Patrick Ryder, a spokesman for the Joint Staff, said the presence of US observation posts in Syria was not discussed during the meeting.
On Wednesday, Joint Staff officials said reports of Gen. Dunford's comments regarding the observation posts were false. They have also paved the way for the formation of a constitutional committee.
Mattis' order followed on Turkish artillery strikes against the SDF in northern Syria, which led those forces to call a temporary halt to the offensive against ISIS near the Iraqi border.
Mr Erdoğan - who considers the Kurdish People's Protection Units (YPG) terrorists - said Turkey would "not make concessions" and preparations for an offensive were almost complete.
Turkish President Recep Tayyip Erdogan accused Bolton of making "a very serious mistake" with the demand.
The killing led the U.S. Senate to vote to suspend U.S. support to the Saudi-led coalition fighting in Yemen, as well as condemn Crown Prince Mohammed bin Salman for the act.
He also doubled down on his recent threats to attack the Kurds.
"We don't think the Turks ought to undertake military action that's not fully coordinated with and agreed to by the United States at a minimum, so they don't endanger our troops", Bolton said Sunday in Israel.
The U.S. has been working with allies for months to figure out what to do with all the suspected ISIS militants in Kurdish custody.
According to the YPG, a 16-year-old boy from the USA identified as Soulay Noah Su was among the alleged terrorists who were trying to carry out an attack on civilians.
"There are different voices within the administration", Cavusoglu said in remarks to a parliamentary committee, according to Turkey's Hurriyet newspaper.
During a visit to the Middle East earlier this week, National Security Advisor John Bolton appeared to walk back Trump's assertions.
"After being so intertwined and engaged with a terrorist organization, it is not that easy to leave that terrorist organization", he added, referring to the YPG.
"America is a force for good in the Middle East", Secretary of State Mike Pompeo said Thursday in Cairo, in an expansive speech in which the top US diplomat repudiated the Obama administration's Middle East policies and accused it of making crucial errors that worsened a string of crises in the region. Iranian officials, Trump said, "can do what they want" in Syria.
Earlier this week, Trump, who has lost his secretary of defense James Mattis and his representative to the anti-Islamic State (IS) Coalition Brett McGurk over his surprise announcement on Syria, seemed to be taking a step back on the decision to withdraw the USA troops when he said the timeline was still being decided.
Answering a question on whether he wants USA troops to stay, President Salih said: "We will need the support of the U.S".