The officials declined to describe the unusual contact between the two governments, and whether there had been an Iranian reply. Senior Obama administration officials have said publicly that Iran would cross a “red line” if it made good on recent threats to close the strait, a strategically crucial waterway connecting the Persian Gulf to the Gulf of Oman, where 16 million barrels of oil — about a fifth of the world’s daily oil trade — flow through every day…

The secret communications channel was chosen to underscore privately to Iran the depth of American concern about rising tensions over the strait, where American naval officials say their biggest fear is that an overzealous Revolutionary Guards naval captain could do something provocative on his own, setting off a larger crisis…

Estimates by naval analysts of how long it could take for American forces to reopen the strait range from a day to several months, but the consensus is that while Iran’s naval forces could inflict damage, they would ultimately be destroyed.

“Their surface fleet would be at the bottom of the ocean, but they could score a lucky hit,” said Michael Connell, the director of the Iranian studies program at the Center for Naval Analysis, a research organization for the Navy and Marine Corps. “An antiship cruise missile could disable a carrier.”