More than a dozen US states say Syrian refugees are no longer welcome due to security fears after the Paris attacks.
Governor Rick Snyder of Michigan said he was suspending the acceptance of new arrivals until after a review.
Alabama, Texas and several other states issued similar statements but a State Department spokesman said the legality of this action was still unclear.
President Barack Obama has urged the US to "step up and do its part" to help those fleeing the civil war.
"Slamming the door in their faces would be a betrayal of our values," he said.
"Our nations can welcome refugees who are desperately seeking safety and ensure our own security. We can and must do both."
The governors' decisions come in the wake of the attacks in Paris which killed 129 people on Friday evening.
Seven of the perpetrators died in the attacks, and one of them is thought to have been a Syrian who entered Europe via Greece with migrants.
Millions of Syrians have fled to neighbouring countries and to Europe, and the US has promised to take about 10,000 Syrian refugees in the next 12 months.
States saying no:
While the state of Alabama has not accepted any Syrian refugees so far, the southern state's governor has said that he "will not place Alabamians at even the slightest possible risk of an attack on our people".
In Michigan, where it has been reported that 200 Syrians have been resettled in the past year, Governor Snyder has said he will suspend the acceptance of new refugees until the US Department of Homeland Security "completes a full review of security clearances and procedures".
At the scene: Suzanne Kianpour, BBC News Washington
US State Department lawyers are investigating whether governors can legally block Syrian refugees from being settled in their states.
While the final ruling has not been made, officials say it appears they can't actually block refugees who come into the country through the resettlement program, but they can make it difficult for the non-profit organisations doing the legwork.
A state can tell an NGO handling a file that they don't want to work with them and the NGO would simply go elsewhere. But uneasiness about the vetting process and fear of a similar attack in the US is growing, especially on Capitol Hill.
The decision to suspend the acceptance of refugees has drawn the ire of some working to resettle them.
"It's vital to keep in mind those who are refugees are fleeing persecution," said Michael Mitchell, with the Lutheran Immigration and Refugee Service, an organisation working to resettle refugees in the US.
But several Republican presidential candidates have said it would be wrong to accept any more. Business mogul Donald Trump described it as "insane".
The three top Democratic candidates have said they want the US to take more than 10,000 Syrian refugees but only after extensive vetting.
On Monday it was reported that a new video released by the so-called Islamic State celebrates the attacks and threatens Washington, DC.
Flags on many public buildings across the US are flying at half-mast as an expression of solidarity with France.