Tensions have since been high, and the Israeli army has carried out several air attacks, killing one fighter and wounding several Palestinians.
Palestinian fighters have launched dozens of home-made rockets into Israel, wounding a handful of people.
Tzipi Livni, the Israeli foreign minister, is scheduled to come to Cairo on Thursday for talks with Mubarak about a new truce.
Abbas also said he and Mubarak agreed that reconciliation talks between Hamas and Fatah, of which he is the leader, should go forward.
"If efforts failed, they have to pick up again because the Palestinian dialogue is a fundamental Arab and Palestinian interest. Also, canceling or violating the truce cannot be accepted and Egypt will do all it can to restore calm," Abbas said.
Commenting on the anti-Egyptian protests in a number of countries (that hold Egypt responsible for the Gaza siege), he said Egypt has been exerting every effort to re-open the border crossing points, describing these allegations as "null and void".
He also said Israel cannot put the responsibility of Gaza on Egypt's shoulders and that any Israeli military action in Gaza is "absolutely unacceptable. But at the same time, we are also against the rockets that are fired, and we'll do our best to stop both these violations."
Regarding setting a date for legislative and presidential elections to be held simultaneously (an issue Hamas rejects), Abbas said they will not happen "until all attempts for dialogue yield no result, because dialogue is the important basis for what we do. If we don't succeed, then we will cross that bridge when we get to it".
Earlier, a Hamas official said the Palestinian group together with other armed factions had agreed to halt the firing of rockets into Israel for 24 hours to allow aid to be delivered to Gaza.
Ayman Taha said they accepted "a calm for a 24-hour period following Egyptian mediation in exchange for the delivery of aid from Egypt".
But on Tuesday evening, the Israeli military said its troops crossed into northern Gaza to shoot three armed Palestinian men as they tried to plant a bomb.
|The despatch of an Egyptian aid convoy may not make much of an impact on Gaza [AFP]
Israeli media said the three died, but Palestinian sources had no immediate word on the incident.
In other news, an Egyptian convoy carrying supplies worth $1m - five lorries carrying 40 tonnes of flour, 20 tonnes of rice and medical equipment - was expected to enter Gaza on Tuesday.
Hossam Zaki, a spokesman for the Egyptian foreign ministry, told the AFP news agency that "we asked both parties to create a suitable atmosphere to allow an Egyptian humanitarian convoy from the Egyptian Red Crescent to enter Gaza in security.
"The two parties dealt with the Egyptian request in a positive manner."
Al Jazeera's correspondent Ayman Mohyeldin, reporting from Gaza, said that even if the aid convoy is allowed to enter the territory, the delivery will not be enough to have any lasting impact on the shortage of supplies.
"The United Nations Relief Agency, which distributes food to around 750,000 Palestinians, says it needs around 100 trucks of wheat to meet the demands of the population," our correspondent said.
"And the World Health Organisation told us that 105 critical life-saving medicines have reached zero level, so they are desperately in need of stock.
"The situation remains very dire and desperate according to international organisations here in Gaza."