Asked whether the meeting eased states’ concerns about accepting refugees from crisis areas such as Syria and Iraq, Mr. Herbert replied, “We still have work to do.”

After Mr. Obama announced his plan last fall to admit up to 10,000 refugees into the U.S. this year, governors of 30 states from Maine to Wyoming publicly asked for the resettlement of Syrian refugees to stop until security concerns and procedures could be addressed. New Hampshire Gov. Maggie Hassan was the lone Democrat in the group.

The resistance to refugees intensified after Islamist terrorist attacks in Paris and San Bernardino, California, which raised fears of Islamic State fighters sneaking into the U.S. as refugees.

It’s not clear whether states could block the federal government’s effort to resettle refugees across the country, but some governors have issued executive orders to state agencies to do what they can to stop refugees from relocating in their states.

Virginia Gov. Terry McAuliffe, a Democrat, conceded that some states “are confused about the process to go through.”

“A lot of issues were raised. We’re continuing to have the dialogue, and [Obama administration officials] promised they would get back to us with additional information as we go forward,” Mr. McAuliffe said.

Mr. Herbert said one of the governors’ concerns is “making sure we have the appropriate information and have the appropriate policy in place.”

“We understand, as governors, one of our first responsibilities is the public safety of the people within the states that we represent,” he said.

White House press secretary Josh Earnest said Monday that the screening of potential refugees is rigorous and that the administration is committed to working with the reluctant governors, but added that accepting refugees “is another way that the United States demonstrates our exceptionalism and the kind of values that make our country great.”