The Jordanian officers did not saluate the Israeli general, which would be normal practice to a superior officer.
The Jordanian officer thought this was quite funny, apparently.
Jordanian and Israeli officers are said to meet regularly in secret to coordinate security along their common frontier.
He said that at least one Jordanian officer began shooting and that the Americans returned fire, killing him.
Early last year in Pristina, a Jordanian officer opened fire with a semiautomatic rifle after an argument, killing another officer and then shooting himself.
The Jordanian officer was unharmed but was evacuated with difficulty as armed Serbs took to the streets.
The Jordanian officer was killed when the guards returned fire.
Officers close to the investigation said that five Jordanian officers were at the gate when the shooting started but that only one opened fire.
Police investigators said they could not comment as to whether the Jordanian officer had deliberately chosen Americans as his target.
The tip about the infiltration came from Jordanian officers, Israeli security officials said.