It has been a week since Turkey began its attack on Syria, and the situation is changing daily, with Russia now trying to mediate in the conflict.

Turkey's offensive in the Kurdish territory of northern Syria, now entering its seventh day, has led tens of thousands of people to flee their homes and caused dozens of deaths.

While the United States has claimed it is withdrawing its remaining 1,000 troops from northeastern Syria, Russia is trying to fill the void left by the US.

Russian President Vladimir Putin spoke with his Turkish counterpart Recep Tayyip Erdogan about Syria by phone and invited him to visit Russia in the next few days.

The Kremlin said late on Tuesday that in their phone conversation, Putin and Erdogan agreed to ensure Syria's territorial integrity.

"Vladimir Putin invited Tayyip Erdogan to come to Russia with a working visit in the coming days. The invitation has been accepted," the Kremlin said in a statement, adding that the leaders also discussed the need to avoid possible conflicts between the Turkish and the Syrian forces.

On Wednesday, Russian Foreign Minister Sergei Lavrov said Moscow will encourage the Syrian government and Kurdish forces to reach an agreement. The Kurdish leadership of the Syrian Democratic Forces, or SDF, announced an alliance with the Syrian government to fight the Turks on Oct 15.

Russia wants to take this opportunity to expand its influence in the Middle East, said Sun Degang, deputy director of the Middle East Studies Institute of Shanghai International Studies University.

"In (US President Donald) Trump's mind, the Middle East is a burden to Washington, not worth spending more energy and military power," he said. "This gives Russia a signal that this area has a vacuum of power that can be filled."

Russia said its military police were patrolling a zone separating Syrian government forces and Turkish troops, in cooperation with Ankara.

With Trump's critics saying that he handed over the fate of US allies and stretches of Syria to Russian control, Washington tried to play down Moscow's role.

"The number of Russians is very, very limited. But it only takes a few Russians with a big Russian flag to get everybody to pay attention," said a senior administration official cited by Agence France-Presse.

Sun also said Russia currently does not have enough power to fully control the situation.

While Russia is trying to mediate among multiple parties, Erdogan still refused to halt Turkey's military offensive in northeast Syria late on Tuesday, despite growing pressure from the US, which has announced limited sanctions on Turkey and will send its Vice-President Mike Pence to Ankara on Wednesday to try and reach a cease-fire deal.

Erdogan said he told Trump that Ankara could "never declare a cease-fire" or negotiate with "terrorists"-referring to the SDF-until they are eliminated from Turkey's borders.

The Syrian Kurdish fighters are viewed by Turkey as terrorists due to their ties with the outlawed Kurdistan Workers' Party, which has waged a 35-year-long battle for autonomy within Turkey.

Xinhua, Reuters, AP and AFP contributed to this story.