CBS News reports the resignation was not accepted by Fatah.
He reportedly also threatened to quit as premier unless he received clear instructions from Fatah on how to proceed with negotiations with Israel.
Hakam Balaawi, a Fatah official, said Abbas quit the Fatah Central Committee. Fatah, headed by Yasser Arafat, has been in turmoil over declaration of a unilateral cease-fire since last month.
Arafat also has been critical over the way Abbas has conducted peace talks.
Another top Fatah official said that Abbas' move might be a ploy aimed at forcing recalcitrant Fatah members to agree to the way he is handling contacts with Israel, adding that Abbas was likely to withdraw his resignation.
Balaawi said, "I hope that Abu Mazen (Abbas) will change his mind and withdraw his resignation soon." Fatah officials were trying to mediate a solution.
Abbas has been Arafat's deputy in Fatah, the largest Palestinian movement, for decades.
Earlier in the day, militants from the radical Palestinian group Islamic Jihad claimed responsibility for a bombing in central Israel that killed a 65-year-old woman, apparently violating last week's cease-fire pledge.
But Islamic Jihad leaders in Gaza disassociated themselves from the bombing, and also said they remain committed to the truce, reports CBS News Correspondent Robert Berger, again showing an internal split over relations with Israel.
Islamic Jihad's political leader in the West Bank, Sheikh Bassam Saadi, said Jenin-based militants probably staged the attack in order to react to Israel's decision not to release prisoners affiliated with the group. But he stressed that "Islamic Jihad... is committed to the (truce) and it remains so today."
And Islamic Jihad's top spokesman in Gaza, Nafez Azzam, also distanced the group from the claim, saying: "We have no knowledge about the claim of responsibility ... and are still committed to this initiative and the truce. ... We stand by our word and our commitments."
Meanwhile, the U.S. Ambassador to Israel Daniel Kurtzer Monday called Abbas (Abu Mazen) a "relatively weak man" who tends to "run away from problems."
According to the Israeli newspaper Haaretz, Kurtzer was speaking Monday evening to about 150 rabbis and Jewish lay leaders in Jerusalem.
While the truce has been accepted by most Palestinian groups, some renegade groups within Yasser Arafat's Fatah faction have rejected the cease-fire. Last week, one of these groups claimed responsibility for shooting dead a Bulgarian construction worker near the West Bank town of Jenin.
"Abu Mazen, we know, is a relatively weak man," who tends to "run away from problems rather than try to solve them," Kurtzer said, adding that the U.S. goal in the Mideast is not to back Abbas but to get rid of Yasser Arafat and see the establishment of "a serious Palestinian constitution that will outlive its incumbent."