But the draft resolution circulated at UN headquarters in New York faces opposition from veto-wielding members of the Council like Russia and China.
The UN says more than 2,200 people have died since protests began in mid-March.
A UN report has also said Mr Assad's regime could be guilty of crimes against humanity.
The draft resolution put forward on Tuesday names President Assad along with 22 members of his ruling circle, as well as Syria's General Intelligence Directorate.
It calls for targeted sanctions against those "responsible for or complicit in ordering, controlling, or otherwise directing, violent repression against the civilian population in Syria," the AFP news agency reports.
But the US ambassador to the UN, Susan Rice, said the resolution contained no military threat, as the Syrian people "have been very clear they don't want any foreign military intervention".
Ms Rice told CNN that Mr Assad "needs to know that he is on a dangerous and immoral course that will have significant consequences for his leadership".'Dialogue needed'
However Russia - a close ally of Mr Assad and which has veto power in the Security Council - has said it sees no need for further action beyond a council statement on 3 August condemning the violence.
China, which also has veto power, said Mr Assad and the Syrian opposition "should seek to peacefully and properly resolve the issue through dialogue and consultations".
"The future of Syria should be decided by Syria itself," said foreign ministry spokesman Ma Zhaoxu in Beijing.
Both China and Russia voted against a UN Human Rights Council resolution on Tuesday calling for an independent investigation into violence by the Syrian government, including allegations of the use of tanks and snipers against unarmed civilians. The motion eventually passed with 33 votes in favour.
The Security Council said the inquiry would establish the facts in Syria and where possible "identify those responsible with a view of ensuring that perpetrators of violations, including those that may constitute crimes against humanity, are held accountable".
Syria's ambassador said the vote had been unbalanced.
Also on Tuesday, the European Union widened its sanctions against Syria, adding 15 people and five institutions to those already targeted.
Syria's protests first erupted in mid-March and have continued despite the crackdown. The demonstrators are demanding the removal of President Assad, whose family have been in power for 40 years.
As well as civilians, human rights groups say 500 soldiers have been killed and thousands arrested since March. The government has blamed the unrest on "armed criminal gangs".
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