Erdogan and Russian President Vladimir Putin reached an agreement Tuesday that would transform the map of northeast Syria, installing their forces along the border and filling the void left by the abrupt withdrawal of American troops.
Kurdish fighters have been given 150 hours from noon on Wednesday to pull back 30km (18 miles) along the border.
The House passed a similar measure last week with only 60 representatives voting against it. McConnell also said they could impose sanctions on Turkey for its invasion of Syria that occurred after USA forces started leaving.
Kurdish forces are also to withdraw from the towns of Manbij and Tal Rifaat, which are situated farther west, near the city of Aleppo.
The new Russia-Turkey arrangement lays a good groundwork for the final settlement of the conflict in northeast Syria.
Following the announcement, President Erdogan warned that the offensive could be re-launched "even more strongly" if Washington should "break its promises" under a truce deal. "Further reports to come later!"
Turkey's goal was to push back the Syrian Kurdish fighters, considered by Turkey to be terrorists.
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Turkey "paused" its offensive last week under a USA -brokered deal which called for Kurdish YPG fighters to withdraw, and then secured Russian support this week for a wider deal requiring the YPG's removal from the whole northeast border.
Ahead of a meeting of NATO defense ministers on Thursday in Brussels, the United States and NATO cautiously welcomed a German proposal for a security zone in northern Syria, though U.S. Ambassador to NATO Kay Bailey Hutchison said Europe should take charge of it and not U.S. forces.
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"Our country, represented by the head of state, once again offers a peaceful agenda, guarantees stability and demonstrates the ability to have a positive impact on the situation in the Middle East region", Zheleznyak said.
The president added that the YPG "does not have a place in the future of Syria" and that he would seek to end its presence in areas of Syria controlled by the Syrian government, which Turkey opposes.
Russian Federation - a crucial ally of Syria's President Bashar al-Assad - has demanded that Turkey respect the country's territorial integrity and Putin will likely seek such commitments from Erdogan.
He also vowed that "YPG Kurds will not be allowed to remain disguised as regime" soldiers of Bashar al Assad.
Turkish top officials on Tuesday hailed Turkey's "historic" deal with Russian Federation on Syria.
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Last week, Turkey agreed to pause its offensive against Kurdish fighters in northern Syria at Washington's behest, giving time for the YPG to clear out of the Turkish-Syrian border region.
The US addressed the offensive, imposing sanctions on their North Atlantic Treaty Organisation ally, while France and Germany restricted arms sales to Turkey.
News agency RIA Novosti quoted the Russian defence ministry as saying military police had met with officials there and were due to start patrolling.
In the early hours of Wednesday, Turkey said it had been informed by the United States that their withdrawal from the border areas had been "completed".
According to the memorandum of understanding reached between the two countries during the almost-seven-hour-long talks, as provided to Al Jazeera by the Turkish foreign ministry, Russian military police and Syrian servicemen will be deployed to northeastern Syria, while Ankara's operation "Peace Spring" will continue in a limited area.
What is the safe zone that Turkey proposes?
European Council President Donald Tusk condemned the incursion and called on Turkey - which is a candidate for EU membership - to pull out troops.
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