BERLIN—Leaders from France, Germany and Italy proposed Monday to step up cooperation in areas such as intelligence sharing and border defense, without giving any sign that they will grow closer economically in response to British voters’ recent decision to leave the EU.

“In the face of Islamist terror and the civil war in Syria, we have to do more for both our internal and external security,” said German Chancellor Angela Merkel, calling for national intelligence services across the EU to share information with each other...