Trump wants to pull 2,000 US troops out of Syria, where Iran and Hezbollah have propped up Assad during Syria's seven-year civil war, which is winding down. "I think they know who their friends are", Bolton said, referring to the Kurds.
Addressing reporters at the White House, the president said: "We are pulling back in Syria".
Meanwhile fighting continues in Syria's Deir ez-Zor province to liberate the final shreds of territory held by ISIS.
Trump's decision last month drew widespread criticism from allies, led to the resignation of U.S. Defense Secretary Jim Mattis and raised fears over clearing the way for a Turkish assault on the Kurdish fighters. The announcement also raised fears that Turkey would take advantage of the U.S. withdrawal and attack the US-allied Syrian Kurdish People's Protection Units, or YPG, which Turkey considers a terrorist group linked to domestic Kurdish insurgents.
The Kurdish-led force battling the remnants of the militant Islamic State group in eastern Syria said on Monday it captured five foreign fighters, including two U.S. and as many Pakistani citizens. While acknowledging meetings last week in Moscow with Russian and Turkish leaders, Bolton said, "My impression is those meetings in Moscow did not go well".
Trump defended his decision on the conviction that other countries should take on the burden of fighting whatever remains of IS, and has made tentative agreements with Turkey's President Recep Erdogan to essentially hand over the job to the Turks.
Even though the resolution still passed with 151 votes in favor, 14 abstentions and only two votes against it - from the U.S. and Israel itself - the move still was met with thunderous applause in Tel Aviv. "Israel will remain forever on the Golan Heights, and the Golan Heights will forever remain in our hands", Netanyahu said at that time.
Trump contended, however, that he did not say the withdrawal would happen immediately and that he is committed to a "slow and coordinated [U.S.] pullout" from Syria. Smith added: "I'm pleased that John Bolton has recognized the national security interest, and that's what we want to have ... not a tweet going "Eh, let's get out of Syria'".
Kurdish groups that fought alongside USA troops against IS are concerned a speedy United States withdrawal from Syria would expose them to attacks by Turkey, which regards them as terrorists.
Responding to Bolton's remarks, Turkish Presidential Spokesman Ibrahim Kalin maintained Turkey's opposition to the YPG, which he described as oppressing Syrian Kurds. He said we are leaving.
In Washington on Sunday, Mr Trump reiterated that the U.S. would be pulling its troops out of Syria but suggested the move might not happen soon.
On Sunday, Israeli Prime Minister Benjamin Netanyahu asked U.S. National Security Advisor John Bolton for Washington's recognition of Israeli sovereignty over the Golan Heights Sunday.
So it appears that at least some US forces will remain.
Bolton also was to convey the message that the United States is "very supportive" of Israeli strikes against Iranian targets in Syria, according to a senior administration official, who was not authorized to publicly discuss Bolton's plans before the meetings and spoke on condition of anonymity.
Speaking during a visit to Israel, Bolton said that certain "objectives" must be achieved before a pullout could take place. "Moreover, unlike the issue of Jerusalem, the Golan wouldn't be part of any future Israel-Palestinian peace agreement, making US recognition of Israeli sovereignty much more straightforward than any issue involving Jerusalem or other territories like the West Bank".