WASHINGTON—The U.S. is preparing to admit at least 10,000 Syrian refugees in the next year, a six-fold increase from the number currently in the country, the White House announced, as President Barack Obama comes under pressure to respond more forcefully to a growing global migrant crisis.

The proposal follows calls from lawmakers and human rights groups for Mr. Obama to expand the number of Syrian refugees accepted into the U.S. beyond the nearly 1,600 that have been allowed into the country since 2011.

The...