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And the Obama administration was grappling with more troubling news from Saudi Arabia, whose leadership expressed public dissatisfaction with U.S. policy in the Middle East, especially toward Syria, Iran and Egypt. The Saudis are reportedly threatening to scale back their longtime partnership with Washington over Mr. Obama’s perceived weakness in dealing with Syria, and for reaching out to Iran’s new president without consulting Riyadh.

“What Obama has never really understood is that you’ve got to treat your allies better than your adversaries,” said Michael Rubin, a specialist on the Middle East at the conservative-leaning American Enterprise Institute.

“He came into office promising to reboot America’s image in the Middle East, and to restore frayed alliances which were allegedly devastated by the Bush administration. What he’s managed to do is ruin relationships which have been carefully cultivated by both Democrats and Republicans for decades, specifically Saudi Arabia and Israel,” Mr. Rubin said.

Secretary of State John F. Kerry tried to downplay the tensions this week, insisting that the U.S. and the Saudis are “on the same page” about the need to resolve the civil war in Syria.

“It is our obligation to work closely with them — as I am doing,” he said after meeting with the Saudi foreign minister.

The fallout with Brazil, France and Germany over U.S. spying is less likely to have a lasting impact, said Joseph Wippl, a former CIA employee and a professor of international relations at Boston University.

“I can’t ever remember that espionage really changed a relationship in a basic way,” Mr. Wippl said. “My view is we don’t have friends or enemies, we just have partners.”

Although Mrs. Merkel has reacted angrily to the reports of U.S. spying, Mr. Wippl said the Obama administration would be foolish not to monitor her closely.

“If we’re not trying to listen in on the conversations of the second-most powerful person in the world, then something is wrong with us,” he said.

Mr. Rubin, and a growing number of critics, say the longer-lasting problem for the U.S. is the belief in some parts of the world that the Obama administration has lost its credibility on foreign-policy matters.

“When the root of the Saudi complaint and the Israeli complaint and the Egyptian complaint is that you don’t consult with us, you’re not treating us like friends, ultimately that suggests a real problem,” Mr. Rubin said. “We’ve managed to convince everyone that the United States is untrustworthy.”

This article is based in part on wire-service reports.