In calls to the UN peace envoy for Syria and to the lead opposition negotiator, US Secretary of State John Kerry yesterday said the priority was "a durable, nationwide cessation."
A truce was called in February between Assad's forces and a coalition of rebels but has since begun to break down, particularly in the divided and besieged city of Aleppo.
This week, Russia and the United States agreed to pressure the parties to hold their fire in the Latakia and Eastern Ghouta regions -- but Aleppo was left out of the deal.
With the peace process hanging by a thread, Kerry is scheduled to fly to Geneva today for talks with UN envoy Staffan de Mistura and the Saudi and Jordanian foreign ministers.
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But first, he used his calls to emphasise that Washington has not conceded the idea Aleppo can be excluded from the ceasefire nor that civilians can be targeted.
In calls to De Mistura and the general coordinator of the Syrian opposition High Negotiations Committee, Riad Hijab, Kerry expressed "deep concern" about Aleppo.
In the calls, Kerry dismissed the Russian and regime claim that the Aleppo strikes are targeting the Al-Nusra Front, a jihadist force that is not party to the ceasefire.
"The secretary made clear that we urged Russia to take steps to stop regime violations, especially its indiscriminate aerial attacks in Aleppo," Kirby said.
"The Assad regime continues to escalate the conflict by predominantly targeting innocent civilians and parties to the cessation of hostilities -- not Nusra, as the regime falsely claims.
"The secretary underscored that the initial efforts to reaffirm the cessation of hostilities in Latakia and Eastern Ghouta are not limited to these two areas and that efforts to renew the cessation must and do include Aleppo."
Once Syria's economic hub, Aleppo has suffered some of the worst fighting in a conflict that has killed more than 270,000 people and displaced millions.