But already there are signs that top UN officials will conclude that there is not enough time to hold direct elections before 30 June -- the date by which Washington wants to create a sovereign Iraqi government.
Brahimi's spokesman Ahmad Fawzi discussed those concerns in an interview with RFE/RL's Radio Farda, just before the UN team headed back to New York. Fawzi said elections before 30 June were doubtful.
"Elections between now and 30 June are highly unlikely,” he said. “It is unlikely that you would be able to organize elections in the current environment for many, many reasons, not least of which, of course, is the security environment."
He said some of the problems include “the presence of militias all over the country, heavily armed militias who might intimidate voters and candidates, the lack of a legal framework, and the lack of a political consensus. So, you have to put all these things in place before you can even start preparing for an election. Between now and 30 June, there is very little time to do so."
Since the UN delegation left Iraq on 14 February, several other key players also have suggested they see the question of direct elections as closed.
Top U.S. administrator in Iraq L. Paul Bremer told a U.S. television news program over the weekend that "there is general agreement that there is not time to hold elections in the timeframe of June. Indeed, I think that will be the conclusion of the UN," he said.
At a weekend conference in Kuwait, the foreign minister for the U.S.-appointed Iraqi Governing Council (IGC) said much the same. Hoshiyar Zebari told reporters: "After the findings of the UN mission in Iraq, and from my understanding, it would be extremely difficult to hold elections before the handover of power on 30 June."
There has been no public comment from the chief proponent of direct elections, Grand Ayatollah Ali al-Sistani. But after meeting with the pre-eminent Shi'a leader last week, Brahimi said the cleric "agrees with me that elections cannot be established unless the appropriate circumstances are provided." Al-Sistani's camp is reported to now have some compromise proposals ready but will not reveal them until after UN Secretary-General Kofi Annan makes his formal recommendations based on Brahimi's report.
David Newton, director of RFE/RL's Iraq Service, says there are signs that al-Sistani might agree to forming a transitional government without direct elections if a firm date were set for a poll soon afterwards.
"If he has a commitment to elections reasonably soon, there might be some flexibility there," Newton said.
Sistani has called for direct elections out of opposition to U.S. plans for a caucus system to choose the members of the sovereign government. A direct vote could assure a dominant voice for Iraq's estimated 60 percent Shi'a majority.
Under a caucus system, representatives are chosen by consensus, usually in small meetings or assemblies. Caucus systems are often used to balance the interests of competing groups -- which in Iraq include Shi'a, Sunni, and Kurds.
As the key players await Annan's assessment, the political debate appears now to be shifting toward how to transfer power to a transitional sovereign government that could rapidly take the country to direct elections.
Newton says much of al-Sistani's motivation to date in insisting on elections prior to the U.S. handover is a desire to secure a major voice for the Shi'a community. But al-Sistani also appears to feel elections are needed to prevent what the cleric says is the U.S. interest in turning over power to Iraqi leaders who formerly opposed deposed President Saddam Hussein from exile and many of whom have close ties to Washington.
"What we hear is that he is particularly concerned that the U.S. will turn power over to exiles who are not representative, in his view, of the Shi'a community, and he says that he only speaks for the Shi'a community. He says that he wants genuine leaders from within the community," Newton said.
Newton says that al-Sistani seems particularly concerned by secular Shi'a political leaders like Ahmad Chalabi of the Iraqi National Congress and Ayad Alawi of the Iraqi National Accord. They and other former leaders in exile are currently key members of the IGC.
Some members of the IGC are now proposing ways to form a transitional sovereign government that would be an alternative to both the U.S.-proposed caucus system and al-Sistani's call for popular polls.
Newton says that the proposals range from expanding the IGC by adding new members to convening some kind of popular assembly to give a transitional regime increased legitimacy.
"In the [Iraqi] Governing Council they are talking about perhaps doubling the size of it or calling a kind of national council, something like the Afghan Loya Jirga. But it is all very vague because I think everyone is waiting for Annan to make his definitive pronouncement," Newton said.
It remains unclear whether the UN might call for delaying the handover date beyond 30 June and -- if it did -- whether the U.S. would accept such a proposal.
Bremer, speaking to CNN over the weekend, said that any new approach "may be different from the caucus plan. It may be a modified caucus plan, it may be some form of partial election. It may be some mechanism of a national conference."
But Bremer added, "We believe 30 June is a date that can be hit and will be hit" for creating a sovereign government.