UN Secretary-General Kofi Annan says he hopes to announce a decision on Iraqi election plans by the end of February.
The UN team are meeting different groups in Iraq
He says a UN team, in Iraq to assess conditions for a planned transfer of power by 30 June, is expected to finish its work in about a week.
The US wants regional meetings to select a new government, arguing that conditions are not right for elections.
But a leader of Iraq's majority Shia community insisted that elections could be held by the 30 June deadline.
He gave the UN team a "scientific study" setting out the case for elections.
Speaking at UN headquarters in New York, Mr Annan said "the work of the team is going extremely well".
"They are reaching out and are open to talk to as many groups as possible... The atmosphere has been good. They have been well received and there have been very good and frank discussions."
He added: "I think they [the team] will be there for a week or so, and I hope to be able to give my decision to the [Iraqi] Governing Council before the end of the month."
The nine-member mission, led by top UN diplomat Lakhdar Brahimi, arrived in Iraq on Saturday.
Mr Brahimi said the UN would work to "help the Iraqi people out of its long ordeal and restore independence and sovereignty."
Current head of the Iraqi Governing Council (IGC) Mohsen Abdul Hamid welcomed the offer of support from the UN, which has not had an official mission in the country since a devastating bomb attack last August prompted the evacuation of staff.
The UN envoys met members of the IGC on Sunday - but Shia leaders were notable by their absence from that meeting.
The team met Shia leaders separately.
Abdul Aziz al-Hakim, leader of the main Shia party, said he gave Mr Brahimi a "scientific study carried out by experts" that proves early elections can be held.
The Shia, the majority in Iraq, have been demanding direct elections, with their spiritual leader Ayatollah Ali al-Sistani leading the call.
US officials point to a precarious security situation, the absence of accurate voter lists and the lack of an updated election law as reasons why an early ballot cannot be held.
They have set 30 June as the deadline for handing back sovereignty.
The BBC's Barbara Plett reports that the Americans hope the UN will convince Ayatollah Sistani to accept at least a version of the coalition's handover plan.
They cannot afford to alienate him, she adds, because without support from the majority Shia community, no transition process will work.