Calling for "urgent international action" to protect civilians, Zeid also slammed what he called an "epic failure of global diplomacy" to end the seven-year-old war.
"The past week has been one of the bloodiest periods of the entire conflict, with wave after wave of deadly airstrikes leading to civilian casualties in areas of Eastern Ghouta and Idlib," Zeid said in a statement.
According to the Geneva-based UN Human Rights Office, at least 277 civilians were killed last week including 230 in air strikes by Syrian government and allied forces, the statement said.
"The no-holds-barred nature of this assault is evidenced by reports that at least nine medical facilities, six of them in Idlib and three in Eastern Ghouta, were hit by air strikes," the statement added.
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"After seven years of paralysis in the Security Council, the situation in Syria is crying out to be referred to the International Criminal Court, as well as for a much more concerted effort by states to bring peace," the statement said.
"The conduct and management of this war has been utterly shameful from the outset, and the failure to end it marks an epic failure of global diplomacy."
The draft resolution proposed by Sweden and Kuwait also calls for an immediate end to all sieges including in Eastern Ghouta.
The Council is expected to begin debate on the resolution Monday.
The position on the resolution of Syria's ally Russia was not yet known.
Moscow has repeatedly blocked action in the council that would put pressure on Syrian President Bashar al-Assad.
Earlier this week the Council failed to back an appeal by UN aid officials for a month-long pause in fighting after Russia rejected the proposal.