BERLIN—Germany should offer passports to young British citizens following the U.K.’s vote to leave the European Union, Vice-Chancellor Sigmar Gabriel said Saturday.

Mr. Gabriel said he would raise the issue of dual-citizenship, which is generally forbidden in Germany for non-EU citizens, in the country’s national elections next year.

“Let’s offer it to the young Britons living in Germany, Italy or France so that they can remain EU citizens,” Mr. Gabriel said at a meeting in Berlin of his center-left Social Democratic Party.

The party took a hard-line stance in the wake of the U.K.’s vote leave the bloc, saying the country should complete the divorce process as quickly as possible to avoid wasting the bandwidth of the remaining EU members.

But on Saturday Mr. Gabriel said young Britons shouldn’t be punished for their country’s vote to exit the EU. “Better said, it was older people in England and Wales who decided” to leave, Mr. Gabriel said.