Abbas spoke to Condoleezza Rice, the US secretary of state, on the phone to discuss the upcoming elections and stressed that Israel must not stop Palestinian candidates from campaigning.
He also demanded that Israeli closure of some West Bank areas should be lifted.
Rice is said to have confirmed Washington's commitment to carry out the elections in eastern Jerusalem.
In a related development, the resistance group Hamas called on the Palestinians to confront the Israeli government's decision to prevent its candidates from campaigning in Jerusalem.
Usama Hamdan, representative of Hamas in Lebanon, said the movement is determined to carry out its electoral campaign in Jerusalem, despite the arrest of ten of its candidates and the closure of its campaign office.
Saeb Erekat, the Palestinian chief negotiator, meanwhile lauded US efforts to ensure the smooth conduct of Palestinian polls.
Doubts over the 25 January polls have been voiced after Israel said it would not allow Hamas to campaign in East Jerusalem.
"The US side is making great efforts, asking Israel to allow Palestinian candidates to carry out electoral campaigns," Erekat told Aljazeera.
He confirmed that all those registered in the electoral committee's list of candidates would participate in the elections.
"We want Israel to give us guarantees not to harass, assassinate or arrest candidates"
Palestinian chief negotiator
"However, there are some limits and conditions on carrying out electoral campaigns in eastern Jerusalem. This is applied on all forces, factions and figures participating in the elections. Therefore, all this fuss by Israel is very strange.
"We want Israel to give us guarantees not to harass, assassinate or arrest candidates. We want closures to be lifted and checkpoints to be removed.
"We have asked the US administration to ease all these procedures for us," Erekat said.