Israeli television quoted Shaul Mofaz, the defence minister, as having told Abbas that unless a crackdown on fighters begins by Sunday, Israel will re-designate two commercial terminals on the Gaza boundary as border crossings, which could choke off Palestinian trade.
There was no immediate comment on Friday's report from officials close to Mofaz, who is running for leadership of the right-wing Likud party in advance of Israeli general elections in March.
Saeb Erikat, the chief Palestinian negotiator, said he was unaware of such a threat but said it would be inconsistent with a US-led "road map" to peaceful Palestinian statehood in the West Bank and Gaza, alongside Israel.
"This should not be acceptable to anyone," Erikat said.
Surge in violence
Israel kills Palestinian
On Saturday, an Israeli naval patrol killed at least one Palestinian man off the shore of Gaza, Palestinian medical officials and the Israeli army said.
An Israeli army spokeswoman said that a naval patrol boat spotted two men swimming from Egypt towards Gaza before dawn and that each had a sack tethered to his leg. The boat opened fire on them after they ignored calls to stop.
Palestinian paramedics said the body of 37-year-old Nazer Farahat was retrieved from the water by local fishermen who were continuing to look for the second man.
Israel's navy has largely blockaded Gaza's coast during much of a five-year-old Palestinian uprising, forcing Palestinian boats to stay close to shore.
Israel left Gaza in September after 38 years of occupation, stirring peace hopes. But violence has continued.
Palestinians are now in control of
the Rafah border crossing
After a Palestinian bomber killed five Israelis on Monday, Mofaz ordered a resumption of air strikes against fighters. Three leading Gaza fighters have since been killed.
Political sources said Israel has also suspended talks with the Palestinians on allowing bus convoys between Gaza and the West Bank, despite a deal brokered last month by Condoleezza Rice, the US secretary of state.
That agreement led to the Palestinians taking control of the main Gaza-Egypt border crossing terminal at Rafah and also called for the bus convoys to start next week, followed by truck convoys in January, using a "safe passage" to the West Bank.
In addition to Rafah, Palestinians in Gaza have two commercial crossings into Israel - Karni and Erez. Though Israel has declared an end to its occupation of Gaza - a claim disputed by Palestinians - it has not formalised border crossings.