A senior Obama administration official said the American message to Egypt had been “that we cannot have this conflict drag on, as it just risks greater threats to civilians.”

If Israel goes back into Gaza, both Egypt and Jordan — the only two Arab countries with peace treaties with Israel — would come under pressure from their people to break off ties, a move that would undoubtedly strengthen Hamas.

But to the relief of Obama administration officials, Mr. Morsi so far has not hinted at such a move, which would threaten the 1979 Camp David peace treaty between Israel and Egypt, a linchpin for stability in the region in Washington’s view. And administration officials say Mr. Morsi has indicated that he will try to calm the situation in Gaza before it worsens.

Whether that effort extends to lobbying for Hamas to crack down on jihadist groups that have been launching attacks on Israel, as Israel would like to see Mr. Morsi do, is not clear. But at the moment, the relative quiet out of Cairo is being viewed in Washington as a positive first step.