Ambulances raced to the area to see if there were any casualties from the blasts.
Earlier on Wednesday, Israel killed a Hamas gunman in its second round of air strikes in as many days on the Gaza Strip, responding to rocket fire at its southern towns that wounded three people.
Separate raids by Israel on Tuesday killed three members of the Islamist group in control of the coastal territory. The Israeli military said the strikes were in response to a total of 60 rockets that have been fired at Israel this past month.
The strikes followed a visit by the Emir of Qatar which broke the isolation of the Hamas leadership of Gaza, dismaying Israel and rival, Western-backed Palestinian leaders.
Witnesses in Gaza said at least 10 rockets had been fired at Israel on Wednesday alone, although Israeli police put the figure at 40. Israeli paramedics said three people sustained shrapnel injuries as a result of the rockets, which also damaged a house, according to Reuters.
Israel kept schools shut in communities near the Gaza border and residents were urged to remain indoors.
Two loud explosions shook Gaza City shortly after the attack that killed the Hamas gunman, witnesses said, adding Israel had apparently targeted militants firing rockets. There were no reported casualties. Israel had no immediate comment.
Defense Minister Ehud Barak noted the explosive device was rigged to a gate on the fence, and said Israel takes the attack “very seriously,” according to AFP.
Hamas officials also reported Israeli tanks firing into Gaza. A military spokesman confirmed that report.
The cross-border violence has escalated since a bomb blast wounded an Israeli officer on patrol on Tuesday. Hamas, which has controlled Gaza since 2007, issued a statement claiming responsibility for that attack, jointly with the Popular Resistance Committees.