Mahmoud said the US-led coalition against Isis was supporting the ground operation from the air, but that American special forces stationed in the area had not entered the city.
Captured by the militants in 2014, Raqa became synonymous with Daesh atrocities including beheadings and public displays of bodies, and also emerged as a hub for planning attacks overseas.
"They are still advising and assisting" the SDF, he said, saying United States troops were accompanying SDF units moving into the city but "the SDF have the lead in the fight".
The citadel sits on a hilltop roughly two kilometers (just over a mile) from the city limits.
The U.N. children's agency warned Friday that the fight for the Islamic State group's stronghold of Raqqa threatens more than 40,000 children, while overnight airstrikes on the city in northern Syria killed more than a dozen people.
"However, in the course of their offensive, the government troops were confronted with counteraction from the aviation of the US-led global coalition".
It added that at least seven other civilians were killed in air strikes late on June 8 and early on June 9.
An estimated 300,000 civilians were believed to have been living under Daesh rule in Raqa, including 80,000 displaced from other parts of Syria.
"Many are caught in the crossfire", UNICEF official Geert Cappelaere said, urging all parties to give safe passage to those who want to leave.
New smart speaker expected as Apple kicks off conference
That device was just a multi-touch feature phone with a few Apple utilities until it became everything with the App Store launch. Is it too late for Apple's speaker to be successful? Samsung and Google already have VR systems centered on their smartphones.
The anti-IS coalition have dropped leaflets on Raqqa, instructing civilians on how to flee the city safely. "So in any areas they see advances, they are using the civilians".
The SDF has scored a series of victories since launching its operation to take Raqa in November, including capturing the strategic town of Tabqa and its adjacent dam in May.
Talal Sillo, a spokesman for the Kurdish-led SDF said Tuesday that the "great battle" had begun.
Along with Mosul in Iraq, Raqa was one of the twin pivots of the self-styled Islamic "caliphate" that Daesh declared almost three years ago.
Iraqi forces backed by the coalition are also battling IS in Mosul, where the jihadists are now confined to a few neighbourhoods.
Meanwhile white phosphorus attacks on Raqqa were reported for second night on Friday.
Surovikin also reportedly criticized the U.S for trying to block Syrian government forces from taking control of the country's southern border, AP reported.
The U.S. has dealt the Syrian government and its allied troops several blows near Tanf close to the border with Jordan, where the U.S. trains Syrian opposition forces.