"We are according Abu Mazin (Abbas) a limited delay to let him decide whether to fight against terrorism," the source said on Tuesday.
"If he does not change the rules of the game, then the Palestinians will pay an enormous price."
Abbas gave orders on Monday to his security services to prevent attacks by resistance groups such as Hamas and is to hold talks in Gaza City on Wednesday with leaders of the factions to persuade them to agree to a new ceasefire.
Hamas spokesman Sami Abu Zuhri told Aljazeera: "The statement issued by the Palestine Liberation Organisation (PLO) on Monday was harmful to our Palestinian people.
"The PLO decisions issued on Monday deprive our people from exercising their right of resistance, justify the Israeli aggression against the Palestinians and may also justify the occupation of our Palestinian land as they call on the Palestinian forces to halt their military resistance," he said.
Israeli army radio reported on Monday that chief of staff Moshe Yaalon had demanded that plans be drawn up for a large-scale land operation in Gaza if attacks continue.
"As the elected president, he must choose whether to remain hostage to the terrorists and do nothing or to decide to act against them."
"The statement issued by the Palestine Liberation Organisation on Monday was harmful to our Palestinian people"
Sami Abu Zuhri,
The official said that Abbas could decide to deploy "30,000 armed men in the areas of the Gaza Strip where the terrorists fire rockets and mortar shells against Israeli territory or the settlements".
Palestinian security sources say they have received orders from Abbas for intensive deployments of security forces around the border crossings in Gaza into Israel.
Abu Zuhri said: "We, Hamas movement, assure you that we will continue our resistance in all the Palestinian lands.
"The PLO has no right to issue such decisions. It does not represent all the Palestinian forces, not even the major ones.
"We will receive Abbas when he arrives in Gaza for talks, particularly about the truce. We will meet Abbas, listen and talk to him with open hearts and ears," he said.
When asked about Hamas' conditions for a truce, Abu Zuhri said: "We have clearly said this is not the suitable time to discuss a truce. We cannot discuss this now, under the circumstances of continuing occupation of our land, aggression of our people, the closure, the destruction, and the killings of women and children.
"The Israeli enemy should start halting all kinds of aggression against the Palestinian people, and all the Palestinians will then decide on what serves their welfare," he added.
Sharon ceased contact with the
PA after six Israelis were killed
Sharon severed contact with the Palestinian Authority after an attack on one of the Gaza crossings on Thursday that left six Israelis dead. The attack was claimed by three factions, including an armed offshoot of Abbas' Fatah organisation.
The aide to Sharon said that if Abbas did crack down, Israel would be prepared to coordinate its plan to withdraw troops and settlers from the Gaza Strip later this year with the Palestinian Authority.