The latest exchange came as Israel warned Hamas it was risking "war" by failing to stop fire balloons being launched across the border.
Egyptian security officials shuttled between the two sides in a bid to end the flare-up, which has seen more than a week of rocket and fire balloon attacks from Gaza and nightly Israeli reprisals.
In Gaza City on Wednesday, demonstrators burned Israeli flags and pictures of Prime Minister Benjamin Netanyahu, in protest at last week's announcement that the United Arab Emirates was establishing relations with Israel.
"The Zionist occupation must be boycotted, and it must be criminalised, rejected, and it must end," senior Hamas official Khalil Al Haya said during the protest.
Fayez Abu Eita, of the rival Fateh movement led by Palestinian President Mahmoud Abbas, also took part in the rally.
Israeli President Reuven Rivlin issued a warning to Hamas during a visit on Tuesday to firefighters in southern Israel who said they were called out to 40 blazes caused by Gaza arson balloons during the day.
"Hamas should know that this is not a game. The time will come when they have to decide... If they want war, they will get war," said Rivlin, whose post is largely ceremonial.
Israel has waged three aggressions on Gaza since 2008.
A Hamas source told AFP the Islamists had held talks with the Egyptian delegation in Gaza on Monday before it left the territory for meetings with the Israelis and the West Bank-based Palestinian Authority.
The Egyptian delegation was expected to return to Gaza after those talks were concluded, the source added.