The six-member Gulf Cooperation Council, of which Saudi Arabia is the most powerful member, expressed "profound worry," the bloc's secretary general, Abdullatif al-Zayani, said in a statement.
He said the law "contravenes the foundations and principles of relations between states, notably sovereign immunity."
The US House of Representatives passed the Justice Against Sponsors of Terrorism Act by a unanimous voice vote on Friday.
The Senate already approved the bill four months ago and it now goes before President Barack Obama.
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The White House has signalled that he is likely to veto the bill but his veto can be overridden by a two-thirds majority of both houses.
Zayani expressed hope that the US administration "will not endorse this law... which will create a grave precedent".
Saudi Arabia is a longstanding US ally but it was also home to 15 of the 19 Al-Qaeda hijackers who carried out the September 11, 2001 attacks on the United States which killed nearly 3,000 people.
The draft law would allow attack survivors and relatives of terrorism victims to pursue cases in US federal court against foreign governments and demand compensation if such governments are proven to bear some responsibility for attacks on US soil.
Two GCC members -- Qatar and the United Arab Emirates -- also issued their own separate statements on Monday criticising the bill.