Another PLO official said Abbas, who heads the Fatah faction in the West Bank, will give a speech later on Thursday explaining his decision not to run.
Saeb Erakat, the chief Palestinian negotiator, said: "At the end of the day it's not the presidency.
"It's the question of the Israeli government continuing settlement activity, fait accompli policies, dictation; and nineteen years after trying to achive a two-state solution, maybe the president has come to his moment of truth, and I think he'll specify this personally."
Abbas decree
Abbas recenty issued a decree announcing that presidential and parliamentary polls would take place on January 24.
But the Hamas-run interior ministry in the Gaza Strip ordered Palestinians in the territory not to take part in the elections.
The interior ministry said in a statement the elections had been called "by figures who do not have the right to declare it".
Nour Odeh, Al Jazeera's correspondent in Ramallah, West Bank, said: "We have confirmation that the Palestian president has informed the top leadership of his Fatah movement that he [Abbas] doesn't intend to seek re-election.
"He wants Fatah to come up with a different candidate and he told the PLO executive committee the same thing this afternoon.
"Both leadership bodies [Fatah and PLO] have rejected this decision. They have urged him to reconsider but he hasn't given them a final answer."
Abbas's decision not to take part in the elections comes days after Hillary Clinton, the US secretary of state, visited Israel to try to kickstart the Israeli-Palestinian peace process which has stalled over Israel's refusal to halt settlement activity in the occupied territories.
Israeli position
Al Jazeera's Jacky Rowland in Jerusalem said: "The official position of the Israeli side is that this is an internal Palestinian matter and, therefore, Israel won't be making any comment at all.
"But in a way it really takes the heat off Binyamin Netanyahu [the Israeli prime minister]. Afterall, it was only under a week ago that the US secretary of state was here again supposedly trying to act as some kind of intermediary to get the whole peace process moving again.
"And very much it's been a position of Netanyahu that Israel has been ready to talk; it's been the Palestinians standing in the way of talks although that might be rather a disingenuous argument. Now he [Netanyahu] can say, 'Look, we're ready to talk but there's no partner on the Palestinian side. So it really does take the heat off the Israelis."
Abbas has of late been facing heavy criticism for defending a decision to delay the endorsement of a UN report on Gaza war crimes at the UN Human Rights Council.
Although the council later passed a resolution adopting the report, Abbas continues to be criticised by Hamas, a rival Palestinian faction in coontrol of the Gaza Strip, which has called on Palestinians to reject his leadership.
The bickering between Fatah and Hamas has derailed a unity deal the two factions have been pushing forward under the auspices of the Egyptian government.
After the storm that was generated by the October 2 delay of the vote on the Gaza war report, Hamas said it had asked Egypt's intelligence chief to put off the reconciliation deal until November.