HELSINKI—Finnish Foreign Minister Erkki Tuomioja said his country has contingency plans for a possible breakup of the euro, marking a rare disclosure from a euro-zone official about the extent of concern over that prospect.

Finland's European affairs minister Alexander Stubb quickly downplayed Mr. Tuomioja's remarks, saying that "every government has various scenarios that they are working with. Our No. 1 scenario is that the euro zone will regain stability and we are working...