In the Syrian capital yesterday the army pressed a major assault to crush rebels around the capital, a monitoring group and activists said.
And in the contested city of Aleppo in the country's north, rebels attempted to advance into western regime-held districts, sparking clashes with government forces, the Syrian Observatory for Human Rights said.
"The army is trying to take over Qaboon, Barzeh, Jubar, Al-Hajar Al-Aswad and Yarmuk," said Observatory director Rami Abdel Rahman, referring to neighbourhoods in the northern, eastern and southern outskirts of the capital.
Kerry met leaders of the Sunni Arab monarchy as part of a regional tour in which he has called for greater support for Syria's rebels but stressed that Washington ultimately wants a political solution that includes all sides.
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Saudi Foreign Minister Prince Saud al-Faisal told Kerry that Assad, a secular leader who belongs to the Alawite sect, an offshoot of Shiite Islam, has waged "genocide" through the more than two-year conflict that has claimed nearly 100,000 lives.
"The regime's illegitimacy eliminates any possibility of it being part of any arrangement or playing any role whatsoever in shaping the present and future," he said.
Faisal voiced dismay at the role of rival Iran, which has poured assistance to Assad to save its main Arab ally. Hezbollah, the Lebanese Shiite Muslim group backed by Iran, is fighting in support of Syrian government forces.
Kerry has previously voiced fears of "ethnic cleansing" in Syria.
President Barack Obama has vowed to step up support for the rebels after concluding that Assad defied warnings and used chemical weapons. But he is cautious about deeper involvement in an increasingly sectarian conflict.