Their talks come a day after Bush and South Korean President Roh Moo-hyun vowed a nuclear-armed North Korea will not be tolerated and agreed talks should be held on a North-South peace treaty, a show of harmony that played down tactical differences.
Bush is eager for unity among the five nations engaged in negotiations with North Korea - the United States, South Korea, Japan, Russia and China - to try to convince Pyongyang to agree to dismantle its nuclear arms programmes.
Aides said Bush, in his fifth meeting with Putin this year, would also discuss Iran's nuclear programme, Russia's effort to gain membership in the World Trade Organisation and Russia's internal situation.
Ties between the US and Russia have at times been strained over Washington's worries that Moscow is backsliding on economic reforms.
Both leaders are in Pusan for the Asia-Pacific Economic Cooperation summit.
Bush (R) began the current tour
in Japan on Tuesday
Before seeing Putin, Bush will meet several leaders of the Association of Southeast Asian Nations, or ASEAN, and will talk about, among other issues, his efforts to pressure Myanmar to make improvements in human rights.
Bush is on a weeklong trip including stops in China and Mongolia. He began the trip in Japan.
Presidential counsellor Dan Bartlett said Bush would keep fighting on the issue, saying the criticism had reached a critical mass and that it "requires a sustained response."
"I think it's not only fair game for the president to correct the record, I think it's his obligation," he said.
N Korean position
North Korea said in September it would disarm in exchange for aid and security guarantees and a light-water nuclear reactor for civilian use.
Bush and Roh have not always agreed on the best way to deal with North Korea, with the US side consistently taking a more hard-line approach and South Korea more inclined to provide incentives to entice the North down the path of peace.
Roh and Chinese President Hu Jintao on Wednesday had called on all six parties involved in the North Korean negotiations to show flexibility.
"I think it's not only fair game for the president to correct the record, I think it's his obligation"
Bush opposes help with the reactor until North Korea disarms. The next round in the six-party talks are to resume in December at the earliest.
Mike Green, Asian affairs director for the White House National Security Council, said he expected allied leaders to start putting down some "concrete ideas" in the next month or so on how to implement the September agreement.
He said Bush and Roh did not talk about specifics and Roh did not ask for a more flexible US position. . "It was a strategic discussion," he said.