Annan announced this month that he would resign Aug. 31 as joint U.N.-Arab League envoy to Syria, after failing to broker a cease-fire as the country descended into civil war.
Brahimi will travel to New York and then Cairo in the coming days. Speaking to the Associated Press by telephone from Paris, Brahimi said: "I realize it's an extremely complicated and very, very difficult mission." He said he hopes military intervention isn't necessary and that talking about a military option is akin to admitting diplomatic failure.
In Washington, White House spokesman Josh Earnest said that Brahimi was a "capable and seasoned diplomat."
Brahimi, 78, who emerged last week as the leading candidate to replace Annan, brings a long record of working in the Arab and Islamic world. He served as Algeria's foreign minister from 1991-93 and joined the United Nations in 1994, where he served in a variety of high-profile posts until he retired in 2005.
As an Arab League envoy, Brahimi helped negotiate the end of Lebanon's civil war.
Several U.N. diplomats, speaking on condition of anonymity because they were not authorized to speak publicly, said Brahimi had delayed taking the job as Syria envoy because he wanted a signal of support from the council. What kind of support Brahimi wanted remains unclear.
Gerard Araud, France's U.N. ambassador and current Security Council president, has called the special envoy post something of an "impossible mission" and said he could understand why someone would take their time before deciding to take it.
Annan said when he announced his resignation on Aug. 2 that the Security Council's divisions prevented the united approach needed to stop the fighting in Syria. Russia and China have used their veto power three times to block strong Western- and Arab-backed action against President Bashar al-Assad's regime.
On Thursday, Araud announced that the Security Council had agreed to end the U.N. military observer mission and back a small new liaison office that will support any future peace efforts. The 15 council members agreed that international efforts to significantly reduce violence and end the Syrian government's use of heavy weapons - conditions set for possibly extending the observer mission - had failed.