While the U.S. military has said it will continue to support the SDF, the United States does not have a presence in Afrin.
Any such operation would be considered an "aggressive act", Mekdad said.
Last week's news that Washington was training the predominantly Kurdish Syrian Democratic Forces (SDF) to form a 30,000-strong "border force" based in Afrin infuriated the Turks. Turkey reacted angrily to the announcement.
Sporadic cross-border artillery exchanges between the two sides have been ongoing at least since August, but increased on Thursday after the U.S. announced it would maintain and train a 30,000-strong Kurdish force in northern and eastern Syria to stabilize the regions against ISIS.
Turkey has repeatedly voiced threats against Afrin, branding the People's Protection Units (YPG) operating in the region as an offshoot of the banned Kurdistan Workers' Party (PKK), classed as a terrorist group by Ankara.
In Afrin, residents took to the streets to protest Turkey's threats, according to photos published by the Kurdish news agency Hawar.
Turkey has been angered by US military support for the Kurdish YPG-dominated Syrian Democratic Forces which spearheaded the fight against Islamic State in Syria, and by an announcement that the United States would stay in Syria to train about 30,000 personnel in the swathe of eastern Syria under SDF control.More news: Instagram Adds 'Last Seen', Here's How You Turn it Off
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People hold flags and carry a banner reading "We condemn and denounce the attacks of the Turkish government against our people in Afrin" during a protest against Turkish attacks on Afrin, in Hasaka, Syria, January 18, 2018.
He called on Russia to support the operation and said Turkey would work with the country "to prevent any mishaps" that could affect Russian forces in the area. It has been unable to stop Israeli airstrikes against suspected Hezbollah targets within Syria.
Turkish leaders were not satisfied.
Washington later said the effort had been mischaracterized and that the US was not creating a border force, but that the coalition would provide security to liberated areas, blocking escape routes for Islamic State militants. "There's been an increasing shelling from Turkey into Afrin", said Dekker.
"The promises made to us over Manbij were not kept".
The incursion, dubbed Operation Olive Branch, is likely to sharpen tensions between Turkey and the United States, which supports and openly arms Kurdish militias fighting ISIS. As a result the army that the Kurds dominate, the Syrian Democratic Forces (SDF), now controls nearly half of Syria's territory.
Russian Federation is Syria's most powerful ally.
Gen. Joseph F. Dunford Jr., chairman of the US Joint Chiefs of Staff, spoke by phone Saturday with his Russian counterpart, Gen. Valery Gerasimov, about a number of issues, including Syria.