He indicated on Friday that the US-led occupation in charge in Iraq had promised to guarantee security for the team to work in the country, from which the world body withdrew three months ago.
"I believe that within a few days the team should be able to travel and start the work," Annan told reporters in Brussels.
The UN pulled its non-Iraqi staff out of Iraq in October because of the deteriorating security situation, after an attack on 19 August on its Baghdad headquarters which killed top envoy Sergio Vieira de Mello and 21 others.
Annan announced in Paris on Tuesday that the UN was to return to Baghdad, after a three-month absence, to conduct the review on holding elections before July.
But he insisted that security remains a concern, and said a decision on sending the team would depend on guarantees by the US-led occupation in Iraq on safety measures for UN staff.
Asked on Friday if security conditions were not good enough, Annan said: "I believe we are in the process of making progress," adding: "The coalition has promised me that it would do its utmost to protect the team."
An advance UN team arrived in Iraq last week to discuss security with the authorities there. On Thursday the UN special envoy to Iraq, Ross Mountain, said in Amman that he expected the team to report soon on whether it is safe.
The UN pulled out from Iraq after
an attack on its headquarters
The UN election study team has been asked for by the US, whose plan to install a government in July appointed by an unelected transitional assembly is facing growing opposition from the majority Shia Muslim community in Iraq.
Annan underlined earlier this week that the UN mission to study the feasibility of elections did not amount to a full return of the United Nations to Iraq.
The mission will be "a team that is going in ... to give advice and then coming out. It is not a return of the United Nations," he said in Brussels on Wednesday.
And he warned: "If no agreement can be reached on that mechanism and a formula for a provisional government then I am very much worried that there will be a continuance of division and conflict."
The June deadline for the transfer of sovereignty is part of a
calendar agreed with the US-appointed interim leadership in November that also sets out a timetable for a constitutional convention and general elections during 2005.
Annan's comments were quickly welcomed by the spiritual leadership of Iraq's Shia Muslim majority, which has spearheaded the campaign for prior elections and which vowed to keep an "open mind" in its dealings with the UN team.
Iraqi Shia oppose selection and
"The UN team should come visit the 'marjaiya'," (Shia Islam's top clerics) said Sheikh Abd al-Mahdi Karbalai, the representative in the holy city of Karbala.
In his sermon at the main weekly Muslim prayers, Karbalai called on the UN team to "base its study on an objective and valid evaluation."
Karbalai later told AFP that if the UN team ruled elections were not feasible before the 30 June handover deadline, the Shia clergy would "insist on a formula closer to elections than designations."
In the nearby holy city of Najaf, prayer leader Sheikh Sadr al-Din Kubanji also urged the United Nations to "make elections its choice."
But although Shia leader Muqtada al-Sadr ignored the issue in his sermon, his representative Sheikh Raid al-Kadhimi lashed out at the United Nations.
"Why should we accept leaving the matter in the hands of the United Nations? Don't the Iraqis have brains and a marjaiya?" he told 2000 worshippers at the capital's Kadhimiya shrine.