BRUSSELS—Negotiators have moved within sight of a deal on terms of Britain’s divorce from the European Union, diplomats from the two sides say, narrowing differences over a key sticking point: How to avoid a physical border between Ireland and Northern Ireland.

While issues remain and a final agreement could still prove elusive, EU and British diplomats close to the talks said London and Brussels appeared close to a deal ahead of a critical summit of EU leaders next week.

If...