The president said he wanted Mr Obama to "hit the ground running"
US President George W Bush has said he and Barack Obama will discuss issues such as the global financial crisis and the war in Iraq "early next week".
In a speech at the White House, Mr Bush congratulated the president-elect and said he would make every effort to ensure a smooth handover on 20 January.
The president-elect is due to hold his first news conference on Friday, after a meeting with his economic advisers.
Meanwhile, Rahm Emanuel has accepted the post of Mr Obama's chief-of-staff.
The Illinois congressman, who was once an adviser to President Bill Clinton, will be responsible for much of the internal management of the new administration.
Mr Emanuel is a tough Washington insider who has been strongly criticised by some Republicans for being too partisan, says the BBC's Jane O'Brien in Washington.
According to ABC News network, Obama campaign strategist David Axelrod has also accepted the job of senior White House adviser - a post previously held by Karl Rove and Dan Bartlett under Mr Bush.
Mr Obama is next expected to appoint a treasury secretary to tackle the country's economic crisis.
Illinois Congressman Rahm Emanuel (L) was an adviser to President Clinton
Likely contenders reportedly include former Treasury Secretary Larry Summers and Paul Volcker, the former head of the Federal Reserve.
There is also speculation as to whether Mr Obama will reach out beyond the Democratic Party and include Republicans in his administration, such as current Defence Secretary Robert Gates.
The BBC's Kevin Connolly in Chicago says the job of White House spokesman may be reserved for Robert Gibbs, a campaign aide whose combination of a jolly personal style and steely on-message discipline captures something of Mr Obama's own persona.
Earlier, Mr Obama was reported to have won North Carolina - which had not voted for a Democrat in more than a generation. The state's 15 electoral votes increase his total to 364. Mr McCain has 162.
WHAT HAPPENS NEXT
9 Dec: Deadline for states to resolve issues such as recounts or challenges
15 Dec: Electoral college electors meet in each state to formally cast their votes
6 Jan: Joint session of Congress to count electoral college votes
Before 20 Jan: Barack Obama and Joe Biden must resign from the Senate
Missouri is now the only state that remains too close to call.
In a televised address to aides on Thursday, President Bush said he would ask government officials "to brief the Obama team on ongoing policy issues, ranging from the financial markets to the war in Iraq".
"I look forward to discussing those issues with the president-elect early next week," he added.
Mr Bush said his administration would do everything it could to ensure a smooth and peaceful transition in advance of Mr Obama's inauguration in 75 days' time, so that he and his team "can hit the ground running".
Mr Bush said the Department of Justice had already arranged security clearance for key members of Mr Obama's transition team, and that the Illinois senator had begun to be given classified intelligence and security briefings.
The president said the transfer came at a challenging time.
Bush sympathises with his staff
"We're in a struggle against violent extremists determined to attack us and they would like nothing more than to exploit this period of change to harm the American people," he said.
Because the US does not have the same culture of a neutral civil service as most democracies, many officials at most levels of government are political appointments, so thousands will leave with the old president and thousands more will arrive with the new one, our correspondent says.
Mr Obama has begun talks about forming his new administration.
He has until inauguration to select senior officials.
The transition team is to be run by John Podesta, a former chief-of-staff to President Bill Clinton, Pete Rouse, who was Mr Obama's Senate chief-of-staff, and his close friend, Valerie Jarrett.
The entire US House of Representatives and a third of US Senate seats were also contested in Tuesday's elections.
The Democrats increased their Senate majority by five seats to 56, including two independents, but fell short of the 60 needed to stop blocking tactics by Republicans. Four seats are too close to call.
The Democrats also increased their majority in the House of Representatives, gaining 19 seats to give a total of 254, leaving the Republicans with 173.
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