The Kerem Shalom crossing, which is used to transfer humanitarian cargo into the Gaza Strip, is likely to remain closed on Monday for an evaluation of the security arrangements, Haaretz said.
There was still no decision by Sunday evening on whether to open the Sufa crossing.
As for the crossing at Nahal Oz, where fuel is transferred to Gaza, Palestinian fighters fired at a fuel truck on Thursday.
Quoting sources, Haaretz reported that Dor Alon, the company operating the terminal, had informed the army that it intended to re-evaluate its role in transfering fuel to the Palestinians in light of the dangers to its staff.
Dor Alon refused to confirm the information provided by the security sources.
Two weeks ago, two Israelis were killed in a Palestinian attack at the terminal.
Israeli political sources said on Sunday that "Hamas is harming the civilians of the Gaza Strip".
The same sources said that about 200 lorries carrying food and medicines cross Kerem Shalom every week on their way to Gaza, in an effort to avoid a humanitarian crisis there.