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Early Thursday, Mr. Netanyahu warned that the Sinai is becoming a “terror zone” after a rocket fired from the territory struck the southern Israeli resort city of Eilat. No injuries were reported.

The prospect of a further deterioration in relations between the two countries would raise difficult questions for Washington, which has given Egypt roughly $2 billion in aid annually since 1979.

“If they no longer respect agreements reached under previous governments, then they’re not a country worthy of our support,” said Rep. Gary L. Ackerman of New York, the ranking Democrat on the House Foreign Affairs Committee’s Middle East and South Asia subcommittee.

But Mr. Ackerman, echoing a now-common school of thought in Washington, told The Times that Mr. Shater’s candidacy might be a positive development given the alternative.

“If I was writing the morning line on who can beat the Salafists, it’s the Muslim Brotherhood,” he said. “And if I have to choose between horrible and not that great, I’ll take not that great.”

Susan Crabtree contributed to this report.