Israeli troops found the remains on Tuesday of the first rocket to be fired from Egypt since the July 3 overthrow of the Islamist government there, a military official said.
Both Israelis and Egyptians reported hearing several explosions in the southern city of Eilat on Thursday, the day after President Mohammed Mursi was toppled from power in Egypt.
Israel detected no signs of any cross-border shooting, but found the remains of a rocket on Tuesday, an official said, speaking on condition of anonymity.
An Israeli military spokesman said the rocket remnant had been discovered in the hills north of Eilat, a resort city on the Red Sea that abuts Egypt to the west and Jordan to the east.
The rocket was the first since the latest bout of unrest in Egypt that has put Israel on edge in part because of an increase of Islamist militancy in the Sinai region since an uprising toppled autocratic president Hosni Mubarak in 2011.
In Egypt, a hardline Salafist Islamist group called Ansar Bayt al-Maqdes, whose name means "Followers of Jerusalem," issued a statement claiming to have fired rockets at Eilat, targeting fuel depots and residential areas.
The group accused Jews of responsibility for Mursi's downfall and said this was why it attacked Israel.
Israel has avoided any show of satisfaction over the ouster of Mursi, an Islamist president who alarmed many in the Jewish state but quickly made clear he would not renege on the two countries' 1979 peace treaty.
But a senior Israeli official voiced hope last week that the appointment of Adli Mansour, a top supreme court judge, as interim Egyptian president would lead to a restoration of contacts with the Cairo government largely frozen under Mursi.
In April, two rockets launched from Egypt's Sinai Peninsula crashed in Eilat, causing no casualties or damage. A similar attack was reported last November.