The television reported on Sunday Abbas telephoned Israeli Defence Minister Shaul Mofaz and informed him the Palestinians would do all they could to find Eliahu Gorel.
The taxi driver's car was found undamaged with the engine still running in the Beit Hanina district of occupied east Jerusalem, some 10 km from Ramallah.
The 61-year-old had disappeared on Friday after taking two passengers from Tel Aviv to a hotel in Jerusalem.
Israel immediately put the blame on the PA and called on the authorities to resolve the problem. "If it turns out to be a kidnapping, it is very serious and we await action from the Palestinian Authority," Foreign Minister Silvan Shalom told public radio.
An Israeli army statement had earlier said that "in light of the security assessments made today, there is a sincere concern that terrorist motives stand behind Gorel's kidnapping".
Police said Gorel telephoned his family from the Ramallah area of the West Bank on Saturday saying he was well, before the call was cut off.
Subsequently, they added, a man "with an Arab accent" telephoned to say Gorel had been kidnapped but no harm would come to him.
There has been no indication yet whether Gorel has been kidnapped.
Israeli police chief Shlomo Aharoniski had earlier said police did not yet know the reason for his disappearance or whether it was linked to the prisoners question.
On Friday, the Palestinian resistance group Hamas threatened to kidnap Israeli soldiers if Israel did not make good on its commitment to release Palestinian prisoners.
The main Palestinian resistance groups have been observing a temporary ceasefire in their struggle against Israeli occupation in exchange for the releases.
The Israeli government has decided to free only 350 of some 8,000 prisoners being held - many without charge or trial.
That number fell far short of Palestinian expectations.
Threat to end truce
Hamas officials warned the PA
not to disarm their group
The resistance groups also threatened on Sunday to end their ceasefire if the PA attempted to confiscate their weapons.
"If the Palestinian Authority tries to confiscate our weapons they will force Islamic Jihad and Hamas to cancel the hudna (truce)," said a joint statement issued in Gaza.
"If the Palestinian Authority tries to confiscate our weapons the responsibility for the results lies with the Palestinian Authority," the statement added.
Meanwhile, Israel arrested a suspected Irish bombmaker in the occupied West Bank, Israeli security sources said.
British and Irish newspapers had earlier reported a search was underway for a former Irish Republican Army (IRA) bombmaker suspected of training Palestinian resistance groups in the West Bank.
Security sources said the man was being interrogated to discover the extent of the network he had developed among Palestinians.