The Palestinian president said on Monday: "Today I received American assurances that the campaigning and the elections will take place in Jerusalem."
Abbas's announcement alleviated some fears that he was planning to call off the vote under pressure from members of his Fatah party, who were concerned that the popular Hamas group would embarrass Fatah at the polls.
Incidentally, on Tuesday Israeli soldiers arrested Emad el-Din Kamil, a leader of the Al-Quds Brigades in Qabatiya township, south of Jenin, Aljazeera's correspondent in Jenin reported.
About 20 military vehicles besieged Kamil inside a house, opening fire at the house before managing to arrest him.
In an address carried live on Palestine TV on Monday, Abbas said he spoke to several US officials, including Condoleezza Rice, the secretary of state who gave him assurances that Israel would allow voting in Jerusalem.
Abbas said he also received a message on Monday from George Bush, the US president.
Earlier on Monday, Israel said it would allow candidates who were not members of resistance groups to campaign in East Jerusalem, but added there had not been a decision yet on whether to allow voting to proceed there.
Israel says only candidates who
are not fighters can campaign
Mark Regev, spokesman for the Israeli Foreign Ministry, said: "We are hopeful that we are close to an understanding.
"We want to reach an agreement under which Palestinians living in Jerusalem who want to vote in the elections can, and at the same time ensure that we don't give legitimacy to any terrorist group."
Sean McCormack, the US State Department spokesman, said two US envoys would travel to the region on Tuesday to smooth out arrangements for the elections, as well as other security issues.
Palestinian leaders had said the elections would have been delayed if voting was banned in East Jerusalem.