Rahm Emanuel, a Democrat congressman from Illinois, has accepted Obama's offer to become White House chief-of-staff.
And there were also reports that Obama was set to appoint David Axelrod, the chief strategist of his election campaign team, as a senior advisor.
White House talks
Obama will make his first visit to the White House as president-elect on Monday afternoon, accompanied by his wife Michelle, said Bush's spokeswoman, Dana Perino.
Bush, who will soon head home to the state of Texas after two terms in office, has asked White House staff to ensure a peaceful transfer of power.
"We face economic challenges that will not pause to let a new president settle in," Bush said.
|Emanuel has accepted the post of White
House chief of staff [EPA]
"This will also be America's first wartime presidential transition in four decades. 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."
The meeting between Bush and Obama will come just days before world leaders gather in Washington for a November 15 summit to discuss the global financial crisis.
An Obama aide said he is set to give his first news conference as president-elect on Friday, after he holds a meeting with his key economic advisers.
Rob Reynolds, Al Jazeera's correspondent in Chicago, said previous transitions to power have not always gone smoothly.
"In the past there have been some real hand-over horror stories. After president Bush took office in 2001, his staff complained that Clinton officials and staff had vandalised some of the offices."
Obama is also set to select some of his economic advisers, including his choice for treasury secretary.
Among those being considered for the role of treasury secretary are Timothy Geithner, president of the Federal Reserve Bank of New York; Lawrence Summers, a former treasury secretary; and Paul Volcker, former federal reserve chairman.
But in a reminder of the tough challenges awaiting Obama, Asian and European markets slumped again on Thursday over renewed economic worries.
Whoever gets the job will be faced with guiding America's $700bn economic bailout package of financial institutions and the regulatory changes that analysts hope will prevent a repeat of the crisis.
Emanuel, who like Obama, is from Chicago, helped mastermind the Democrats' capture of the House of Representatives from the Republicans two years ago.
Reynolds said Emanuel is seen "as a tough politician with strong links to Israel".
John Mercurio, the political editor of Hotline, an online briefing service, told Al Jazeera that Emanuel's reflected Obama's confidence in his abilities.
"When crucual decisions are to be made, he [Obama] chooses people with experience, and those closesly connected to the political establishment," he said.
"It is also worth remembering that the media has been extremely favourable towards Obama during the election campaign. Now he has been elected, and if his goals and promises do not appear to be fulfilled, he will face a lot more scrutiny."
Obama has already launched a transition team that is working fast to fill the next administration's economic and homeland security teams, according to one of the Democratic sources.
Heading up that team are Valerie Jarrett, a close friend of Obama; Pete Rouse, his US senate chief of staff; and John Podesta, former chief of staff to Bill Clinton.
Felipe Calderon, Mexico's president, congratulated Obama via telephone on Thursday.
Calderon's office said Obama pledged continued US support for Mexico's fight against organised crime and drug trafficking.
A statement from the Mexican president's office said Obama told Calderon he was "conscious of the difficulty of the battle" and offered "decisive" US support.
Nearly 4,000 people have died this year alone in drug violence in Mexico, most of them members of warring drug trafficking organisations.