His remarks before Palestinian lawmakers in Ramallah on Monday sparked widespread condemnation and reignited accusations that Abbas is a Holocaust denier.
The proposed statement would express the council's "serious concern" about Abbas' remarks, which "included vile anti-Semitic slurs and baseless conspiracy theories, and do not serve the interests of the Palestinian people or peace in the Middle East".
"I would like to assure everyone that it was not my intention to do so, and to reiterate my full respect for the Jewish faith, as well as other monotheistic faiths", he said.
Abbas added that the PA has a "long held condemnation of the Holocaust as the most heinous crime in history" as well as "anti-Semitism in all its forms".
"From the 11th century until the Holocaust that took place in Germany, those Jews - who moved to Western and Eastern Europe - were subjected to a massacre every 10 to 15 years", he said.
Israeli officials had no immediate comment on Abbas' statement of apology on Friday.
The United States has twice blocked draft statements at the council expressing concern about the violence in Gaza, in which almost 50 people have been killed by Israeli forces.
He also said that Zionism and the founding of Israel was a European colonial project, saying "history tells us there is no basis for the Jewish homeland".
Mr Abbas and Palestinian officials have refused to cooperate with the administration of US President Donald Trump, who they perceive to be the most pro-Israeli American leader since its creation in 1948.
He then cited "three books" written by Jews as evidence that 'hostility against Jews is not because of their religion, but rather their social function, ' adding he meant 'their social function related to banks and interest'.
Israeli Prime Minister Benjamin Netanyahu said Wednesday that the comments were anti-Semitic.
'Such rhetoric will only play into the hands of those who do not want a two-state solution, which President Abbas has repeatedly advocated'.
Mr Trump ordered the relocation of the United States embassy from Tel Aviv to Jerusalem, which will take place on May 15, on the anniversary of what the Palestinians call Nakba Day, the commemoration of their "day of catasrophe". The Palestinians seek East Jerusalem as the capital of any future state.