Ukraine’s leaders believe Russia is preparing a broad attack on their country, European Commission President Jean-Claude Juncker said Tuesday as he returned from a summit in Kiev that coincided with fresh violence in Ukraine’s east.

In an interview with a small group of reporters on his flight back from the Ukrainian capital, Mr. Juncker said he had advised the country’s President Petro Poroshenko to ease tensions by avoiding comments on a possible bid to join the North Atlantic Treaty Organization. His advice, he acknowledged,...