Ireland’s central bank Wednesday said it is working with the country’s banks to make sure they can weather the fallout from a potential U.K. exit from the European Union, an eventuality that it says would damage Ireland’s economic recovery.

The U.K. is set to vote June 23 on whether to leave or remain in the bloc, an option that is popularly known as “Brexit.” While economists are divided on how much harm a vote to leave the EU would inflict on the U.K. economy, they are agreed that Ireland would suffer most among other European...