He added, however, that Olmert did not say what he meant by "almost."
Abbas said he would be meeting with Olmert again on October 27, perhaps for the last time.
According to the Palestinian president, the talks he held recently with Syrian President Bashar Assad were sincere.
"Assad did not express his reservation over the things I presented him with in terms of the internal Palestinian issue, and he presented me with the entire situation regarding the Israel- Syria conflict."
Abbas denied reports that he had conveyed a message to the Syrian leader from US President George W. Bush.
The Palestinian leader confirmed Monday morning's reports that Egypt had issued invitations to the Palestinian organizations to attend talks in Cairo next month, based on early elections, a unity government, and the reestablishment of the PLO institutions and Palestinian security organizations.
He stressed that his stance remains that the Palestinians must not accept temporary agreements or "shelf agreements" with Israel. He also said he would not accept a solution of temporary borders.
The negotiations between Israel and the Palestinians focus on six main issues: Jerusalem, the borders, the settlements, the refugees, security arrangements and water.
On Sunday, Abbas criticized the Israelis for failing to protect Palestinian farmers during the olive harvest season against the settlers' violence.
He complained about the situation to Israel and the international community, saying that "if Israel fails to take control of the situation, it means it's not serious about attaining peace."