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.
Obama said in a statement that he and his wife Michelle "look forward to meeting with President Bush and the first lady".
"I thank him for reaching out in the spirit of bipartisanship that will be required to meet the many challenges we face as a nation," he said.
But as both incoming Democrats and outgoing Republicans struck a tone of civility, Obama's transition chief signaled that the president-elect could end 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 destruction of human life, and plans 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 health care, on stem cell research," he 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that he wanted all the Bush executive orders reviewed, and decide which ones should be kept, and which ones should be repealed, and which ones should be amended".
Incoming presidents typically visit the White House to get acquainted with their future home and be briefed by the current president and first lady.