Syrian rebels have ruled out respecting the safe zones, though Russia, Turkey, Iran, and the Syrian government will respect them. De-escalations zones are to be established in some of Syria's regions as the continuation of Russian-initiated ceasefire agreement a year ago.
In Moscow, Putin met with his security council to discuss US-Russian relations in the context of Lavrov's meeting with Trump, Kremlin spokesman Dmitry Peskov said.
The uproar seemed certain to complicate Lavrov's mission in search of USA support for a Russian plan to create safe zones in Syria.
"We had a very, very good meeting with Mr Lavrov", Trump told reporters after the talks.
Russian Federation is making great efforts to convince the U.S.to accept Iranian role in a plan for foreign troops to police safe zones in Syria as a step toward ending the six-year war.
The United Nations Special Envoy for Syria, Staffan de Mistura, said on Thursday that next week's UN-facilitated talks in Geneva will benefit from the dynamics of the negotiations which began in Astana, Kazakhstan, earlier this month, which saw an agreement between Russia, Iran and Turkey on "de-escalation zones" in war-torn Syria.More news: Madrid Open: Andy Murray beaten by Borna Coric in third round
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"Today, we have a common understanding that, as active players in the diplomatic process regarding Syria, we are going to pursue these contacts together and with other key countries, especially those in the region", he said.
The agreement calls for the creation of four "de-escalation zones" to shore up a ceasefire, ban flights and allow for humanitarian aid deliveries.
Tillerson and Lavrov's meeting in Alaska comes 150 years after Washington purchased the USA state from Moscow.
Several ceasefires have been agreed on since Syria's conflict broke out in 2011, but they have failed to permanently stem the fighting. Trump said during his election campaign that he sought closer ties with Russian Federation but tensions grew after United States air strikes against a Syrian airfield in April in response to a chemical weapons attack that Washington blamed on Assad.
President Trump's ongoing outreach to Kremlin officials will not prevent him from holding Russian Federation accountable for its "disruptive behavior" in the Middle East and elsewhere, White House national security adviser H.R. McMaster said Friday.