* Turkey, rebel allies capture outskirts of al-Bab
* City left to Turkish control in understanding with Russia-source
* Turkish foreign minister says Raqqa is next target
Syrian rebels backed by the Turkish military have captured the outskirts of the Islamic State-held city of al-Bab in northern Syria, the Turkish government and rebel sources said on Wednesday.
The advance threatens an important IS stronghold, whose fall would deepen Turkish influence in an area of northern Syria where it has created a de facto buffer zone.
Syrian government forces have also advanced on al-Bab from the south, bringing them into close proximity with their Turkish and rebel enemies in one of the most complex battlefields of the six-year-old conflict.
But Turkey said international coordination was under way to prevent clashes with the Syrian forces.
"The al-Bab operation must be completed immediately in the period ahead ... In recent days our special forces and the Free Syrian Army (rebels) have made serious progress," Turkish Foreign Minister Mevlut Cavusoglu told a news conference.
In a sign of Turkish momentum and confidence, the government said its next target would be the Syrian city of Raqqa, de facto capital of the embattled Islamic State group which has also been partly dislodged from its Iraqi stronghold of Mosul.
Al-Bab has been a major target of a Turkish offensive launched in northern Syria last August to drive IS away from the border and prevent further gains by U.S.-backed Kurdish militia that are also fighting the jihadist group. The city is just 30 km (20 miles) from the Turkish border.
A Free Syrian Army rebel commander speaking to Reuters from southeastern outskirts of al-Bab said Syrian government warplanes and helicopters were visible to the west of his position, saying there was now an "indirect frontline" between the sides.
But an official in a military alliance backing Syrian President Bashar al-Assad said the city was being left to Turkish control, in what appeared to be part of a de facto deal with Russia, Assad's most powerful ally.
Turkish Prime Minister Binali Yildirim said clashes with the Syrian forces had been avoided.
"As a result of coordination between coalition forces, the Turkish air force and Russia, necessary measures are being taken to prevent any unpleasant incidents or clashes," Yildirim said.
Assad has been backed in the war by the Russian air force and an array of Iranian-backed Shi'ite militias. The Syrian army advance towards al-Bab is aimed at preventing deeper Turkish advances and safeguarding the city of Aleppo, 50 km (30 miles) to the southwest.
"Progress is fast"
The Turkish military said in a statement that 58 Islamic State militants had been killed in air strikes, artillery fire and clashes. Two Turkish soldiers had been killed and 15 slightly wounded. The advancing forces had captured strategic hilltops around al-Bab, the army said.
A Syrian rebel fighter reached by Reuters said he was speaking from inside al-Bab where Islamic State lines were "collapsing".
"Praise God, the progress is fast," he said. "The operation is continuing."
The Syrian Observatory for Human Rights, a UK-based organisation that reports on the war, cautioned that it was not yet clear if Islamic State had collapsed entirely in the city. It said at least six people had been killed and 12 more wounded in the latest shelling there.
The organisation says Turkish bombardment has killed scores of people since December. Turkey says it has been careful to avoid civilian casualties.
Islamic State is being fought by three separate military alliances in northern Syria, including the U.S.-backed Syrian Democratic Forces which incorporate the Kurdish YPG militia.
U.S. support for the YPG has angered Turkey, which views it as an extension of a Kurdish militia that is waging an insurgency in Turkey.
A spokesman for Turkish President Tayyip Erdogan said Turkey had presented a detailed plan to drive Islamic State out of Raqqa and discussions on the issue were under way.
Spokesman Ibrahim Kalin told broadcaster NTV there had been better coordination with the U.S.-led coalition on air strikes in the last 10 days and Ankara's priority was to establish a safe zone between the Syrian towns of Azaz and Jarablus, which are just over the border.
The safe zone is an important goal for Ankara because it would mean that civilians displaced by the conflict could be provided for in Syria, rather than crossing into Turkey.
Turkish sources said Erdogan and U.S. President Donald Trump agreed in a phone call overnight to act jointly against Islamic State in al-Bab and Raqqa.
The White House said in a statement that Trump spoke about the two countries' "shared commitment to combating terrorism in all its forms" and welcomed Turkey's contributions to the fight against Islamic State, but it gave no further details.
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