Bush, whose sagging popularity helped to propel Obama and the Democrats to election victory, has said he will do "all he can" to help Obama in his transition.
"He can count on my complete co-operation as he makes his transition to the White House," Bush, who will hand over the White House to Obama on January 20, said.
The two men also discussed issues such as Iraq and the global economic crisis during their meeting.
Al Jazeera's Rob Reynolds said the meeting was all about the transfer of power at a perilous time - with the economy in crisis, two wars and several domestic and international problems on the horizon.
"Both teams want to get it right and are anxious to ensure a smooth and easy transfer so the Obama administration can hit the ground running from day one," he said.
In a farewell speech at the White House last week, Bush told more than 1,000 aides that they would need to work with Obama's team to bring them up to speed on critical matters.
However, Obama's aides have said he is planning to reverse a number of current White House policies.
Dan Perino, a Bush spokesman, said after the meeting that the talks had been "good, constructive, relaxed and friendly".
|Obama and Bush discussed Iraq and
the economy [AFP]
Obama's spokesman said that the president-elect welcomed Bush's commitment to ensuring a "smooth" transition of power.
In a statment in advance of the meeting, Obama said: "I thank him [Bush] for reaching out in the spirit of bipartisanship that will be required to meet the many challenges we face as a nation," he said.
His remarks were in contrast to comments while on the presidential campaign trail in which he castigated Bush for his "failed policies".
But as both incoming Democrats and outgoing Republicans struck a tone of civility, Obama's transition chief signalled that the president-elect could end several key Bush policies, including a ban on embryonic stem cell research and moves to open new areas to oil-drilling.
The move could signal a swift change of course after eight years under the Bush administration, even as aides stressed Obama's bipartisan aims and predicted the new cabinet could contain familiar faces.
"On stem cell research, on a number of areas, you see the Bush administration, even today, moving aggressively to do things that I think are probably not in the interest of the country," John Podesta, Obama's transition chief who also served as White House chief of staff under Bill Clinton, said on the US Fox news channel.
The Bush administration moved to limit stem cell research as the president views it as the destruction of human life and he also plans, controversially, to authorise oil and gas-drilling in the western state of Utah.
The incoming administration was reviewing "virtually every agency to see where we can move forward, whether that's on energy transformation, on improving healthcare, on stem cell research," Podesta said.
But Podesta said that he would not "preview decisions that he [Obama] has yet to make".
Podesta pointed out that "as a candidate, senator Obama said that he wanted all the Bush executive orders reviewed, and decide which ones should be kept, and which ones should be repealed, and which ones should be amended".