BRUSSELS—The reintroduction of borders in Europe would destroy the euro and the bloc’s single market, European Commission President Jean-Claude Juncker said in the starkest warning yet about the collapse of border-free travel due to the migration crisis.

Speaking in a news conference Friday, Mr. Juncker said the bloc hasn't “really found any lasting solutions to the refugee crisis.” Over 1 million migrants made their way into Europe last year and in the past few months, several countries, including Germany, have reintroduced...