Asked whether a US withdrawal would not take place until Turkey guaranteed that Kurdish fighters would be safe, Bolton said: "Basically, that's right".
Bolton plans to tell Israeli leaders, including Prime Minister Benjamin Netanyahu, that there is no timetable for an American military pullout from Syria. After holding the unnamed man as an "enemy combatant" on a military base in Iraq for months, the USA planned to release him first to Saudi Arabia, then back into Syria under "safe conditions" in the area where he was captured.
President Donald Trump's abrupt decision to announce a U.S. pull-out from Syria left many questions, chiefly whether Kurdish fighters operating in northern Syria would now be targeted by their long-time enemy Turkey.
On Thursday, Netanyahu said the United States "is acting against Iran at the economic level and we here in Israel are acting against Iran at the military level".
The decision was made after a phone call with Turkish President Recep Tayyip Erdogan, who had been threatening a military operation against the USA -backed Kurds.
He said there are "a lot of options" in the event of another chemical weapons attack like the one that killed dozens in April 2017, saying the next US response would be "more telling" than the last.
"The timetable flows from the policy decisions that we need to implement", he noted.
When asked whether Bolton's comments would affect that timeline, a senior Administration official said that "there is no specific timeline for the withdrawal of USA troops from Syria, and reports to the contrary are false".
"I think they know who their friends are", he said.
"One of the goals of Turkey's struggle against the PKK and its Syrian offshoots is to save the Kurds from the tyranny and oppression of this terror group".
Mr Bolton, who will travel to Turkey on Monday, said the United States would talk to Turkey to find out what its objectives and capabilities were.
He said the withdrawal would take place in northeastern Syria, while some forces may remain to the south at the al-Tanf garrison as part of efforts to counter Iran's presence.
Joining Bolton in Turkey will be the chairman of the Joint Chiefs of Staff, Gen. Joseph Dunford.
In comments carried by the official Anadolu news agency, Ibrahim Kalin said the Kurdish fighters oppressed Syrian Kurds and pursued a separatist agenda under the guise of fighting IS.
Bolton said the US has asked its Kurdish allies to "stand fast now" and refrain from seeking protection from Russian Federation or Syrian President Bashar al-Assad's government.
Bolton visited the Western Wall earlier in the day. He watched a virtual reality tour of the historic site and dined there with his Israeli equivalent, as well as with US Ambassador to Israel David Friedman and Israel's Ambassador to the US, Ron Dermer.
Trump had previously stated that the US' sole mission in Syria - the defeat of Islamic State (IS, formerly ISIS/ISL) - had been completed.