Ms. Merkel made a snap decision that sent shock waves around Europe: Throw Germany’s doors open. Bypassing Europe’s asylum rules and skeptical members of her government, she ordered trains to carry the migrants to Munich.

Her aides couldn’t reach her coalition partner, Bavaria’s premier Mr. Seehofer. He, like Mr. Orban, wanted to stop the migrants; the two men became Ms. Merkel’s most outspoken adversaries. Mr. Seehofer declined to be interviewed.

As Germans greeted refugees in Munich with sweets, toys and hugs, Mr. Orban told Ms. Merkel by phone her decision undermined the fight against illegal immigration and lured migrants to Europe, aides to each say. He lambasted German and Austrian volunteers who drove into Hungary to give Syrians a lift: “Legally they are human traffickers. Is that what you want?”

He told her Hungary was fencing off its southern border. If all EU countries did the same, he said, the crisis would end. “The Hungarian solution,” he said, “is the only solution.”