Israel and the Palestinians are discussing the potential transfer of security control over the West Bank city of Hebron to Fatah security forces in an effort to bolster Mahmoud Abbas.
Mr Abbas's presidential term is due to end on January 9. In the absence of new elections, he will be replaced by Abdel Aziz Dweik, a senior Hamas official and speaker of the Palestinian Legislative Council, currently in an Israeli prison.
Israel hopes that Mr Abbas will stay in the post, but is concerned that if he does not reach an understanding with Hamas over an extension of his term, violent clashes could break out in the West Bank between Fatah and Hamas.
Mr Olmert and Mr Abbas met in Jerusalem on Tuesday to discuss the ongoing negotiations.
Hamas violently took over the Gaza Strip in June of last year. In unrest this week, Hamas clashed with the infamous Doghmush clan from Rafah which runs the Army of Islam and was involved in the 2006 capture of Israeli soldier Gilad Shalit. Eleven people, including a one-year-old baby, were killed.
Fearing a similar Hamas takeover in the West Bank, the IDF Central Command has begun preparing for the possibility that violence will break out in the territories in January if Mr Abbas refuses to step down.
In addition, Israel is also discussing the possibility of expanding Palestinian security control in the West Bank to cities like Hebron, Tulkarm and Kalkilya. Hebron, in particular, has a strong Hamas presence and the transfer of security to the Palestinians would enable them to crack down on it.
Since May, under a pilot programme overseen by the US, the PA has been enforcing security in the city of Jenin. Viewed by Israel as a relative success, the objective was to allow the PA to tighten its control over the West Bank and to ease friction with the IDF which also operates there.
Israeli government spokesman Mark Regev said that Israel was open to the idea.
"We of course would like to see Palestinian security deployed in other parts of the West Bank," he said, "and them taking control and dealing with lawlessness and terror."
